Unrestricted hartree-fock molecular orbital calculationson transition-metal complexes : a detailed study on TiF 3-6de Laat, F.L.M.A.H.

DOI:10.6100/IR162488

Published: 01/01/1968

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Citation for published version (APA):de Laat, F. L. M. A. H. (1968). Unrestricted hartree-fock molecular orbital calculations on transition-metalcomplexes : a detailed study on TiF 3-6 Eindhoven: Technische Hogeschool Eindhoven DOI: 10.6100/IR162488

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•

• -· • •

• --. ~ -UNRESTRICTED HARTREE-FOCK MOLECULAR

ORBITAL CALCULATIONS ON t -1 II TRANSITION-METAL COMPLEXES I

I

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UNRESTRICTED HARTREE-FOCK MOLECULAR ORBITAL CALCULATIONS ON

TRANSITION-METAL COMPLEXES

F.LMAH. DE LAAT

UNRESTRICTED HARTREE-FOCK MOLECULAR ORBITAL CALCULATIONS ON

TRANSITION-METAL COMPLEXES A DETAILED STUDY ON TiF ~-

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HO­GESCHOOL TE EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, DR. IR. A.A.TH.M. VAN TRIER, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG

5 NOVEMBER 1968 TE 16 UUR

DOOR

FRANCISCUS LAMBERTUS MARTINUS ARNOLDUS HENRICL'S

DELAAT

GEBOREN TE NUENEN

(;REVE OFFSET NV h!NDHOVEN

Dit proefschrift is goedgekeurd door de promo tors

Prof. Dr. G.C.A. Schuit

en

Prof. Dr. Ir. P. Ros

AAN MIJN OUDERS

DANKBETUIGING

Bet onderzoek, in dit proefschrift besahreven, k~am tot

stand met de financi~le steun van: het Hoogewerff Fonds, de

Koninklijke Shell en de Stiahting Saheikundig Onderzoek Neder­

land (ZWO), waarvoor ik hier mijn oprechte dank wil uitspreken.

Bet Eindhovens Hogeschool Fonds dank ik voor de financi~le

steun, waardoor ik in staat werd gesteld deel te nemen in 1984

aan het "NATO Advanced Study Institute in Theoretical Chemistry"

te Frascatie (ItaZilt) en in 1967 aan de "Summer Sahool in Theo­

retical Chemistry" te Oxford (Engeland), heiden o.l.v. Prof. C.A. Coulson.

In het bijzonder gaat mijn dank uit naar het Rekencentrum

van de Technisahe Hogeschool Eindhoven voor het veelvuldig ter

beschikking stellen van de EL-XB computer. Zander deze hulp zou dit onderzoek in deze vorm onmogelijk zijn geweest,

Tenslotte wil ik bedanken de heer W.H.J. Kuipers voor de

aorrectie van de Engelse tekst en de heer W. van Herpen voor het vervaardigen van de tekeningen.

TABLE OF CONTENTS

INTRODUCTION • • • •

1.1 Historical review 1.2 Outline of the present work

References

2 HARTREE-FOCK METHODS

. . . .

2.1 Conventional Hartree-Fock scheme.

2. 1 • 1 General theory • • • • • • •

2.1.2 The Hartree-Fock equations • •

2.2 Unrestricted Hartree~Fock scheme

2. 2. 1 General theory • • • • • • 2.2.2 The LCAO-MO approximation

2.3 Projected-unrestricted Hartree-Fock scheme 2.4 Symmetry orbitals ••••

2.5 Slater-type orbitals •

References • • • • • •

3 SINGLE ANNIHILATION FOR A SINGLE DETERMINANT

WAVE-FUNCTION • • • • • • • • • • • • • • • ;:

3.1 Average expectation value of the S -operator

• " • 9

• 9

12

14

17

17

17

20

22

22

24

25

26

30

32

3.1.1 <S2

> before single annihilation ••••

34

34

34

36

37

3.1.2 <82

> after single annihilation •••

3.2 Total electronic energy • • • • •••••

3.3 Charge-density and spin-density functions

References • •

'· 37

39

4 MOLECULAR INTEGRALS WITH SLATER-TYPE FUNCTIONS • • 40

4 • 1 General concepts • • • • • • • • • • 40

4.1 .1 The spheroidal coordinates • • 41 4.1.2 The V 1 (t,·r)-functions. • • • 43 n mp 4.1. 3 The Gaunt coefficients • • 45

4. 2 One-centre integrals • • • • • 49

4. 3 Two-centre integrals • • • • • 52 4.3.1 Two-centre one-electron integrals 52

4.3.2 Two-centre two-electron integrals 55

4.4 Three-centre one-electron integrals • • • 61

4.5 Approximation of three-centre and four-centre

two-electron integrals • • • • • • • • • • • • 63

4.6 Relation between integrals with real and

complex functions • • • • • • • • • • • • • • • • • 67 . 4.7 Description of a real orbital in a rotated coordinate

system • • • References ••

68 • • • • • 70

5 A STUDY OF THE APPROXIMATIONS IN AN UNRESTRICTED HARTREE-FOCK CALCULATION ON TiF~- • • • 5,1 Survey of the computation scheme • • 5.2 Selection of basis sets for Ti and F ••••• 5,3 The treatment of the core-electrons 5.4 Influence of the three-centre and four-centre

two-electron integrals on both the total electronic energy and the 1 ODq-parameter • • • •

72

72

76 eo

84

References ·. • • • • • • • · • • • • • • • • 89

6 SOME COMPUTED QUANTITIES OF TiF~- FOR VARIOUS BASIS SETS

7

8

AND AT VARIOUS METAL-LIGAND DISTANCES· • • • • • 90

6.1 General introduction • • ~ • • • • • • • 90 6.2 computed quantities with the unprojected single

determinant wave-function • • • • • · • • 99 6.2.1 Total electronic energy of the ground state

2T2 and first excited state 2E • • • • 99 g g 6.2.2 Crystal-field splitting parameter (10Dq) • 104

6. 2. 3 Orbital populations · • • • • • • • • • • • 1 07 6.2.4 Testing the Wolfsberg-Helmholz approximation • 109

6.3 Computed quantities before and after spin projection 109 . 2

6.3.1 Average expectation value of the S -operator 109 6.3.2 Charge-density and spin-density functions ••• 111

References • • · • • • · • • • • • • • • • · · •

DISCUSSION References

CONCLUSIONS

LIST OF PRINCIPAL SYMBOLS SUMMARY SAMENVATTING

LEVENSBESCHRIJVING

• 115

116

121

123

127 131 133

135

1 INTRODUCTION

In this chapter a historical review is given of the calcu­lations carried out in the last decades in order to obtain nu­merical data for some properties of the first-row transition­metal complexes. This survey does not pretend to be complete, but mainly refers to an arbitrary number of non-empirical and semi-empirical calculations representative for the progress in the theoretical analysis of the electronic structure of com­plexes. Next, the limitations in these treatments are quoted and the main features of the investigation in this thesis are discussed.

1.1 HISTORICAL REVIEW

Most of the interesting features of the physics and chemis­try of the first-row transition-metal ion complexes are related to the ap~itting of the energy levels of the 3d-orbitals of a transition-metal ion (central ion) under the influence of a

"arysta~-fie~d" caused by the ligands. In oatahedraZ complexes the 3d-orbitals appear to split into two sets: one set of 3d­

orbitals (dx2-y2' dz2) pointing toward the ligands, the other set of 3d-orbitals (d , d , d ) being located between the xy xz yz ligands. An elegant description of this phenomenon can be given by using group theory. The two sets of 3d-orbitals classified within the group theory according to the irreducible representa­tions e and t 2 are further denoted by e and t 2 -orbitals.

g g g g In the special case of a system with a single 3d-electron,

the energy of the system with the electron in a e -orbital will g

be above that with the electron in a t 2g-orbital. The energy

difference is called the "arystaZ-fie~d spLitting" parameter 11

or 10Dq. 1 This quantity can be evaluated experimentally f·rom the position of the crystal-field absorption band in the electronic

spectrum of the complex.

The first attempts to evaluate the parameter 10Dq without using empirical data were made as early as 1939 by Van Vleck2 and Polder. 3 The case chosen by Van Vleck was a central cr3+ion

9

surrounded octahedrally by six water molecules1 Polder studied a

tetragonal surrounded cr2+ion. They used a point-ahaPge or

point-dipoZe approximation for the (negative) ligands and their

calculations gave the correct sign for 10Dq. To some extent

this result is obvious. However, when Kleiner4 extended the com­

putations by using a deZooaZised model (Slater-orbitals 5 ) for

the ligands instead of point-charges, he obtained the wrong sign

for 10Dq, The reason was that in his model the positive nuclear

charge of the ligand attracted the e -electrons more than the g

ligand-electrons repelled them.

tained by Tanabe and Sugano6

mechanical exchange integrals.

The correct sign was again ob-

who also included the quantum-

Sugano and Shulman' made a detailed computation for the oc­

tahedral NiF:- complex at a metal-ligand distance R = 4.014 R which is equal to the Ni-F distance in the K

3NiF

3 crystal. Like

all authors mentioned before, they started from an ionia model

for the complex but allowed the metal~functions to mix with li­

gand functions and permitted different wave functions (composed

of Slater-type orbitals) for the spin up (a) and spin down (6)

electrons (spin-unPestrioted method8). The Ni 2+ electrons that

have been taken into account are the 3d-electrons only. In spite

of the great number of simplifications and approximations ap­

plied in their calculation, they obtained a crystal-field split­

ting parameter 10Dq = 6,350 cm-1 which is in reasonably good

agreement with the experimental value (7,250 cm-1).

But as has been pointed out by Watson and Freeman9and Sim~-'

nek and SroubeklO their method appears to be theoretically in-

correct, since in accordance to the arguments of Watson and

Freeman the ("unpaired") bonding orbitals must be considered in­

stead of the (unpaired) anti-bonding orbitals to characterise 4-the covalency in the NiF6 complex. The value of 10Dq according

-1 to Watson and Freeman's method was only 2,815 em and 2,760

cm-1 dependent on the choice of the method of approximation.

An attempt by Sugano and Tanabe 1 l to elucidate the discrep­

ancy between the result of Sugano 7 and that of Watson9 failed.

O:ffenhartz 1 2 calculated the 10Dq- parameter for NiF~- using the

ionic Hamiltonian model and obtained a value of 4,040 cm-1 •

10

In contrast to this set of calculations for the NiF:- com­

plex, all carried out with Slater-type orbitals (multi-centre

basis set), Ellisl 3 and Ros 1 ~ used a one-centre basis set, 15

i.e. all wave functions of the complex were described in the

same coordinate system. They performed a self-consistent-field 4-(SCF) calculation on NiF6 and included nearly all electrons of

the complex. They obtained a value of 10,800 cm-1 for 10Dq, which is above the experimental value.

Richardsonl6 recently made a set of (spin-restricted) cal­

culations for the octahedral first-row transition-metal fluo­rides and described these systems with Slater functions for the

fluorine and Slater-type functions 17 ' 18 for the central-metal

ion. Also in this work approximations have been applied to de­crease the computation time. His calculations for TiF~- yield values for 10Dq which were quite near to the experimental value,

dependent on the choice of the basis set. Some more or less sat­

isfactory results were obtained for the other fluorides.

Fenske et al.l9 carried out a calculation for the transi­

tion-metal hexafluorides and adapted some parameters to obtain the correct 10Dq. Analogously, the octahedral and tetrahedral transition-metal chlorides 20 were also considered. However, the method by which they determine the value of 10Dq is questionable

as will be shown later.

At the same time others tried to evaluate the crystal-field

splitting parameter by means o~ semi-empiriaaZ methods (see, for example, ref. 21,22), i.e. certain integrals for the potential and kinetic energy of the ions in question are approximated by 'the vaZenae state ionisation energies or potentiaZs (VSIE or VSIP) of the separated ions. These quantities can be obtained

from the tables of Moore. 23 Moreover, one uses in the semi-em­pirical method the Wolfsberg-Helmholz 2 ~ or Ballhausen-Gray25

approximation. Out of this set of computations we shall quote: a calculation on

2+ 25 3- 2- 3-.VO(H20)5 by Ballhausen and Gray, on TiF6 , VF6 and VF6 by Bedon et al., 21 ' 26 on chloroammine complexes of co3+ by Yeranos and Hasman, 27 on NiCl!- by Valenti and Dahl,2a on Fe(CN):- and

Co(CN)~- by Alexander and Gray, 29 on TiF~- by Fenske30 and a

11

general paper by Basch et al. 22 for the octahedral and tetrahe­

dral transition-metal complexes. Other semi-empirical calculations using some non-empirical

quantities instead of VSIE's or VSIP's are: on TiCl~- by Oleari 39 2+ )2+ 32 et al., on Cu(NH3 ) 6 . and Cu(H2o 6 by Roos and on some chlo-

rides of Cu by Ros and Schuit.33 ·

Ciullo et al.3 4 have worked out a general method for the transition-metal complexes which uses empirical parameters as well as exact values for the representation of the repulsion

between two electrons, althoug~ up.to now no application for any special case has been reported.

1.2 OUTLINE OF THE PRESENT WORK

As appears from section 1.1 there has been no calculation on the first-row transition-metal complexes giving us a good in­sight into the reliability of the various approximations ap­plied. Moreover, a lot of authors either evaluate only the crys­tal-field splitting parameter 10Dq or use the experimental value of 10Dq as a basis for determining some auxiliary parameters in their calculations.

A correct description of the electronic structure of tran­

sition-metal complexes (crystals) becomes more and more desira­ble in order to obtain a better insight into various properties and quantities such as: the stability of a complex (crystal),

the crystal-field splitting parameter 10Dq, the charge transfer bands, the total energy as a function of the metal-ligand dis­tance,the ionisation energy, the charge-density and spin-density functions, the hyperfine interaction, direct and super exchange interactions.

Hence this thesis describes a general method yielding re­sults for the total energy, the 10Dq, the orbital populations, the average expectation value of the s2-operator, the total en­ergy as a function of the internuclear distance, and the charge­density and spin-density functions of a complex. This general method will henceforth be indicated by "spin and symmetr>y-un­

restricted Hartree-Fock (UHF) method", in which a linear combi-

12

nation of Slater-type atomic orbitals (LCAO) will be used to

construct the molecular orbitals (MO) .·• To obtain in this method

a self-consistent-field solution an iteration procedure must be followed, which was not done in most of the non-empirical meth­ods7'9'11•12 quoted in section 1.1.

The spin and symmetry-unrestricted Hartree-Fock scheme will I 3-be applied on ad -system: the octahedral TiF6 complex. In this

calculation a single determinant wave function will be used. The influence on the results of the following points will be

discussed in relation to this more or less arbitrarily chosen

complex: (1) approximation or neglection of various integrals occurring

in the computation method~ (2) the iteration proceduret (3} choice of the basis set implying the number of basis func­

tions together with their orbital exponentst (4} the electronic configuration of the complex~ (5) the metal-ligand distance.

The UHF-method yields a wave function which is generally not an eigenfunction of the s 2-operator (no pure spin state), However, a simplified spin-projection technique (single annihi­lation) discussed by Amos et al. 3 5•36 and Sutcliffe 37 will, in

their opinion, correct the wave function into a relatively good eigenfunction. They used this projection method for aromatic

radicals and the NH2 free radical respectively. To get an idea about the effect of the single annihilation on the single determinant wave function of a transition-metal com­plex, some properties (the average expectation value of s2

, the charge-density and spin-density functions) will be evaluated be­fore and after spin projection.

The complete method implying the definitions of all terms used before has been worked out in chapters 2 and 3, the latter mainly dealing with the spin-projection method. All types of integrals occurring in the calculation are discuss-

* These terms will be explained in the following chapters.

13

ed in chapter 4, where the accessory expressions will be derived as well.

Chapter 5 deals with the computation scheme and a detailed cal­culation on the octahedral TiF~- complex. Moreover, the effect

of approximation or neglection of integrals on the following re­sults will be considered: the crystal-field splitting parameter

10Dq and the total energy of a complex. In chapter 6 the influ­ence of the iteration procedure, the basis set and of the metal­

ligand distance is dealt with.

The discussion of all results obtained, the comparison with ex­isting calculations as well as ~ number of conclusions and pos­sible extensions of the computation method developed can be found in chapters 7 and 8.

Concerning the experimental data in the literature for TiF~- we refer to the work by Siegel 38 in which the Ti-F dis­tance in TiF3 was found to be 1,97 i. Bedon et al. 2 1 observed the absorption spectra of NaK2TiF6 , Na2KTiF6 and (NH4 ) 3TiF6 sol­

ids in KCl and KBr pellets and found values of 10Dq for a hypo­thetical octahedral symmetry of 17,500 cm-1 , 17,450 cm-1 and 17,060 cm-1 respectively.

The ab initio calculations on the

the Electrologica XS

spin and symmetry-unrestricted Hartree-Fock

octahedral TiF~- complex were performed on (EL-X8) computer (high-speed memory: 32,000

wordSJ drum: 524,000 words) of the Computer Centre of the Tech­nological University at Eindhoven. The complete set of computer programmes for the evaluation of

the great number of integrals (about 85,000 for each calcula­tion) as well as the SCF-computer programme have been written in ALGOL 60.~ These computer programmes 4 0 as well as the integral values 41 used in the investigation are not included in this thesis.

REFERENCES

l, Dunn, T.M,, McClure, D.S., and Pearson, R,G., Some Aspects of Crystal Field Theory, Harper and Row, New York, Evanston and London, and John Weatherhill, Inc., Tokyo (1965), ch.l.

14

2. Van Vleck, J.H., J.Chem.Phys. 7, 72 (1939). 3. Polder, D., Physica 9, 709 (1942). 4. Kleiner, W.H., J.Chem.Phys. 20, 1784 (1952). 5. Slater, J.C., Phys.Rev. 36, 57 (1930). 6. Tanabe. Y., and Sugano, S:, J.Phys,Soc. Japan !!• 864

(1956). 7. Sugano. S., and Shulman, R.G., Phys,Rev. 130. 517 (1963), 8. Pople, J.A., and Nesbet, R.K •• J.Chem,Phys:-22, 571 L

(1954). --9. Watson, R.E., and Freeman, A.J., Phys.Rev. 134, 1526

(1964). -10. Simanek, E •• and Sroubek, z., Phys.Status Solidi 4. 251

(I 964) • -11. Sugano, s., and Tanabe, Y., J.Phys,Soc. Japan!£, 1155

(1965). 12. Offenhartz, P.O., J.Chem.Phys. 47, 2951 (1967). 13, Ellis, D.E •• MIT Ph.D.Thesis (1966). 14. Ros. P •• (private communication). 15, Ellis. D., and Ros, P., MIT Quarterly Progress Report,

Solid-State and Molecular Theory Group 58, 42 (1965); 59, 51 (1966).

16. Richardson, J.W., (private communication). 17. Richardson, J.W., Nieuwpoort. W.C., Powell, R.R •• and

Edgell. W.F., J,Chem.Phys. 36• 1057 (1962). 18, Richardson, J.W. Powell, R.R., and Nieuwpoort, w.c .•

J.Chem,Phys. 38, 796 (1963), 19. Fenske, R.F., Caulton, K.G., Radtke, D.D., and Sweeney, c.c.,

Inorg.Chem, 5, 951 (1966); 5, 960 (1966). 20. Fenske, R.F., and-Radtke, D.D., Inorg.Chem. 7, 479 (1968). 21. Bedon, H.D •• Horner, S.M., and Tyree Jr •• S.Y., Inorg,Chem.

3. 647 (1964). 22. Basch~ H.,Viste, A., and Gray. H.B., J.Chem.Phys. 44, 10

(1966). --23. Moore, C.E., Atomic Energy Levels, Circular of the Nat.Bur,

Std, No. 467 (1949), Vols. I,II, and III. 24. Wolfsberg, M,, and Helmholz, L., J,Chem.Phys. 20, 837 (1952). 25. Ballhausen, C.J., and Gray, H.B., Inorg.Chem. J: Ill (1962). 26. Bedon, H.D., Hatfield, W.E., Horner, S.M., and-Tyree Jr.,

S,Y., Inorg.Chem. 4, 743 (1965), 27. Yeranos, W.A., and Hasman, D.A., Z.Naturforschg, 22a. 170

(1967). -28, Valenti, V., and Dahl, J.P., Acta Chem.Scand. 20, 2387

(1966). --

29. Alexander, J,J., and Gray, H.B., Coordin.Chem.Rev. ~. 29 (1967).

30. Fenske, R.F., Inorg.Chem. ~. 33 (1965). 31. Oleari, L •• Tondello, E •• Di Sipio, L., and De Michelis, G.,

Coordin.Chem.Rev. 2, 45 (1967). · 32. Roos, B., Acta Chem.Scand, 20, 1673 (1966). 33, Ros, P., and Schuit, G,C.A.~Theoret.chim. Acta (Berl.) ~. 1

(1966) • 34. Ciullo, G., Furlani, c., and Sgamellotti. A., Coordin.Chem.

Rev. 2 • I 5 (l 9 6 7) • 35. Amos, A.T.~ and Hall. G,G .• Proc,Roy.Soc. (London) 263A, 483

(1961). 36. Amos, A.T., Snyder. L.C., J,Chem.Phys • .!.!_, 1773 (1964).

15

37. Sutcliff*• B.T., J,Chem.Phys. 39, 3322 (1963). 38, Siegel, s., Acta Cryst, 9, 684~1956). 39. Backus, J,W,, et al., Numerische Mathematik 4, 420 (1963). 40. DeLaat, F,L.M.A.H., Complete Set of Computer Programmes for

Unrestricted Rartree-Fock Calculations (ALGOL 60), Technische Hogeschool Eindhoven (1968), (unpublished),

41. DeLaat, F.L.M,A,H., Integral Values of Unrestricted Rartree-Fock Calculations on TiFg-, Technische Hogeschool Eind­hoven (1968}, (unpublished}.

16

2 HARTREE-FOCK METHODS

This chapter starts with a brief review of some basic con­cepts followed by the conventional Hartree-Fock method and the accessory Hartree-Fock equations. Some restrictions in this method are rejected and the then resulting unrestricted Hartree­Fock method discussed. Next, the Hartree-Fock equations are transformed into a pseudo-eigenvalue problem by choosing an ex­pansion for the space part of the one-electron functions. This general case will lead to the method of linear combination of atomic orbitals, which is used in the molecular orbital method. An iteration scheme for obtaining a self-consistent solution for the eigenvectors is elucidated.

In connection with the evaluation of spin properties, the projected unrestricted Hartree-Fock method is quoted. A possibility to reduce the pseudo-eigenvalue problem with the help of the symmetry-adapted orbitals will be briefly outlined. ~n the last paragraph we consider the Slater-type function which has been chosen for the description of the one-electron wave function,

2.1 CONVENTIONAL HARTREE-FOCK SCHEME

2.1.1 General theory

In this paragraph a brief survey will be given of the basic concepts of the Hartree-Fock (HF) method; a complete treatment

can be found in the papers of· refs. 1-6.

For a many-electron system (atom, molecule or crystal) we have the Schr~dinger equation H~ = E~, where H is the Hamilto~ nian operator and ~ the wave function or many-electron wave function of the system. If ~ is an eigenfunction of H,the eigen­value E represents the energy of the system. Generally, the Hamiltonian operator contains all kinds of elec­trostatic and magnetic interaction terms such as a term repre­senting the repulsion of the nuclei, the influence of an exter­nal field, the spin-orbit coupling. The Hamiltonian ope~ato~ H with only the electrostatic interac­tions in an N-electron system with fixed nuclei is assumed to be

17

of the form:t

H (1 I 2 I ••• ,N) (2 .1)

where k 1 l7 a,B,y: electron and nuclear indices respectively;

Ak Laplace operator; Z the nuclear charge of nucleus a,a, and y a,!l,y

respectively; the distance between the electron k and nu­cleus Yl

the distance between the electrons k and 1,

and the distance between nuclei a and !l re­spectively.

The operators in the Hamiltonian H can be divided into one­and two-electron operators h(k) and h(kl) respectively. The one-

:...1 electron operators are -~Ak and rky with the physical interpre-

tation of kinetic energy of electron k, and the interaction of electron k with nucleus y respectively. A two-electron operator

-I is rkl representing the interaction of the electrons k and 1.

The Hartree-Fock approximate wave function ~ for a N-elec­tron system is assumed to be a (normalised) anti-symmetrised

produat of N (orthonormal) one-electron wave functions, which can be denoted in a determinantal form, i.e. the single Slater determinant:

'!'(1,2, ••• ,N)

1(11 (1) I/J2(1) ••• t1JN(1)

w1 (2) 1/> 2 (2) ••• 1/JN(2) ( 2. 2)

In the one-electron wave function wi (k) i refers to the function indices, k to the particle indices.

t We use atomic units: e = m = h/2• • I.

18

~he Pauli principle requires:

(1) two electrons cannot occupy identical orbitals1

(2) the wave function should be anti-symmetric in the electrons.

The function of equation (2.2) satisfies both conditions.

The totaL energy of a many-electron system which is repre­

sented by the total wave function ~ is given by:

(2.3)

since ~ (eq. 2.2) is a normaLised function, i.e.

H* ~ d-r = <~I~> = (2.4)

The integration T is taken over all space and spin coordinates

of the N-electrons, and for the integrals the bracket-notation

of Dirac 7 is used. The asterisk indicates the complex conjugate

of the total wave function ~.

The assumption that the set {~i} is orthonormal

<w.l~.> = o(i,j) l. J

(2. 5)

with o(i,j) the Kronecker delta function, is no contraint for

the general solution, since the tjli's are linearly independent

and there consequently always is an orthogonal transformation8

allowing the transformed set of wave functions to form an ortho­

normal set. This transformation does not affect the expectation

values of the total wave function~ (see ref. 9).

Now, we can reduce the expression for the total energy of a N­electron system by substituting equation (2.2) in (2.3), using the orthonormality of set {ljli} (eq. 2.5):

N E L. <w.<t>lh<I>Iw.<I>>

l. l. l. N + ~ L <1jl.(l)ljl.(2)jh(I2)(1-Pl2)1tjl.(l)ljl.(2)>

i,j l. J l. J (2.6)

where P12 indicates the permutation of electrons 1 and 2.

19

2.1.2 The Hartree-Fock equations

In the Hartree-Fock method, the total energy E is to be

made stationary with respect to any infinitesimal variation of

the one-electron function ~i(k) subject to the orthonormality of

set {Jjl.}. This va'l'iational p't'inaiple yields after some algebraic l.

manipulations the following Ha'l't'l'ee-Foak equations (for details,

see ref. 9,1 0) :

(2.7)

The eigenvalues ~ 1 •s being the Hartree-Fock one-electron ener­

gies. The Hartree-Fock operator (or Hartree-Fock Hamiltonian) F in

these equations is given by (cf. ref. 9):

F (1)

G(1)

J. (1) J

K. (1} J

h(1) + G(1)

~ J.<1>- ~ x.C1) J J J J

<lji. (2) ih(12) !~. (2)> J J

<ljoj (2) ihC12JP 12 J!joj (2)>

where h (1): one-electron operator;

G (1): total electronic interaction operator;

Jj(1): Coulomb operator; K. (1): exchange operator.

J

(2.8a)

(2.Bb)

(2.8c}

(2.8d)

The general procedure for solving the HF equations is iterative

and called the Hartree-Fock self-aonsistent;...field (SCF) method. This subject will be discussed in detail in section 2.2.

The one-electron function ljoi{k) can be written as a product

of a space part (o'l'bital) and a spin part since the total Hamil­tonian and consequently the HF Hamiltonian (in our case) does not contain spin terms:

n1 (s) is an eigenfunction

eigenvalue of szi: ± ~.

into equations (2.8a-d)

(2.9)

of the spin operator szi with possible After substitution of equation (2.9)

and application of the orthonormality

properties of the spin functions ni(s), the energy expression

20

{2.6) becomes for a system'with closed-shell structure (see for the definition Roothaan9):

EHF = 2}: H. + 1: (2J .. -K .. ) i ~ i. j ~J ~J

= 1: (Hi+E:i} with i,j = 1, ••• ,N/2 (2 .1 0 ) i

where H. = <<j>.(1llh (1ll<f>.(1}> (2.10a) ~ ~ ~

Jij = «~>· <1> !J. <1> I<~>· <1l> (2.10b) ~ J ~

K .. = <<j>. (1) !K. (1) I<~>· (1)> (2.10c) ~J ~ J ~

For open-shell systems a similar energy expression has been

worked out by Roothaan. 1 0

The conventional HF method described in this paragraph has

some restrictions: { 1 ) the method is spin-~estriated, * 01. i.e. for the set {<f>i} and

{<f>~} holds <4>~14>~> = o{i,j). For the spin-unrest~icted ~ ~ J '

method the orbitals 4>~ may be different from 4>~1 ~ ~

(2) the method is symmet~y-restricted (it consists of symmetry-

restricted orbitals), i.e. the orbital 4>i is a basis func­

tion of an irreducible representation of the symmetry group

of the system.

For symmetry-unrestricted orbitals, 4>i can be any function

satisfying the one-electron HF equation for that system7

{3) both the restricted (conventional) and the unrestricted HF method still contains correlation erro~s (cf. 11,12). How­

ever, the properties which will be investigated in this

thesis are hardly sensitive for these errors.

As a consequence of the constraints (1) and (2), the restricted HF method will give a poor treatment of systems with unpaired

electrons. A disadvantage of the unrestricted HF method is the fact that the total wave function will generally be no eigen­function of the total spin-operator s 2

• However, this disadvan-

•4>~. 4>~ indicate the spatial part of a one-electron wave func-~ ~

tion with a-and 6-spin respectively.

21

tage can be eliminated for the greater part by a spin projection

technique. This subject will be studied in section 2.3.

2.2 UNRESTRICTED HARTREE-FOCK SCHEME

2.2.1 General theory

The spin-unrestricted Hartree-Fock method2 '3' 4 '13 allows

different orbitals for electrons with different spins and there

exists a one-to-one correspondence between the electrons and the

one-electron wave functions. Where we speak of the unrestricted

Hartree-Fock (UHF) method we' mean the spin- and symmetry un­

restricted Hartree-Fock method.

The N-electron wave function in the form of a single deter­

minant with p orbitals 4~ occupied by electrons with a-spin, and l.

q orbitals ¢~ occupied by electrons with S-spin is for the UHF l.

method (denoted in a brief notation) :

'fUHF(1,2, ••• ,N) = (Nl)-j •

• d et { ¢ ~ ( 1 ) a ( 1 ) , ••• ,<I>; ( p) a ( p) ~ .P ~ ( p+ 1 ) S ( p+ 1 ) , ••• , <1>! ( N) S ( N) } ( 2 • 11 )

The abbreviation "det" stands for the Slater-determinant form

(cf.eq.2,2).

The total energy is in this general method, suitable for open

and closed shell systems, analogous to equation (2.10):

a+S a+S a S

a S a+S

L H. + ~ L J .. - ~( L + L )K .• i l. i,j l.J i,j i,j l.J

(2.12)"'

L1 L and L indicate summation over a-, o-, and all occupied or-

bitals respectively. For the meaning of the integrals H~, J .. L l.J

and K .. see equation (2.10a-c). Conform to paragraph 2.1.2 the l.J

Hartree-Fock equations can be deduced for the set {¢~} as well l.

as for the set {¢~}. 1

Solution of the Hartree-Fock equations for complexes is

* The differences between the factors occurring in equation (2. 10) and those in (2.12) are caused by the fact that the summa­tion is in the first instant taken over the orbitals (N/2) and in the second over the electrons (N).

22

such a difficult mathematical problem that it is still out of

the question. An approximation often used in this case, is the

expansion method, 2 in which an orbital ~i can always be expanded

in terms of some complete set of basis functions xt which are

assumed to be normalised. This approach is written as follows:

~~ I Ct Ct xtcti x.c.

~ t - -~

~~ fl x.c~ (2 013)

~ ~ xtcti - -~ t

in which x = (x 1 , ••• ,xn) a row vector and £i = (Cli'''''Cni) a

column vector.

By substituting these equations in the

and following the method of Roothaan for

Hartree-Fock equations

each set {~?l and ~~~}, ~ ~

it is easy to

pr>obZems:

show that this gives rise to the pseudo-eige>Z?''llu<J

(2 .14)

where the matrices Fa= H +Get= H + J- Ka, pfl = H + Gfl H + J

- Kfl and S (overlap matrix) defined by their elements are:

s <x (1) lx (1)> (2 .14a) rs r r

H <x (1) I h(1) lx (1) > (2 .14b) rs r s

J I ( p +Q ) < X ( 1 ) X ( 2 ) I Y' ~ ~ I X ( 1 ) X ( 2 ) ' (2 .14c) rs t,u tu tu r u s t

KCI L Ptu<xr(1)xu(2) lr>~~lxt(1)x 8 (2)> (2. 14d) rs t,u

Kfl rs ~ Qtu<xr(1)xu(2) lr>~~lxt(1)xs(2)

t,u (2. 14e)

P and Q, density matrices for the electrons with ~- and B-spin

respectively, are defined by:

( 2. 15)

The pseudo-eigenvalue problem can be solved by the method

described by Wilkinson1 4 ' 1 5'l6 and a complete computer pro­

gramme can be found in ref. 17.

The total electronic energy becomes with the density matri­

ces P and Q:

tr(PH} + tr(QH) + ~tr(PGa} + ~tr(QGB) a S

~ {tr(PH) + tr(QH) + L e~ + L e~} i 1 i 1

where tr is the trace of the matrix.

2.2.2 The LCAO-MO approximation

(2.16 )

(2.16a)

The choice of a complete basis set in the expansion method

(eq. 2.13) will in many cases be impossible from a practical

point of view (computer capacity and computer time). One usually

takes the set of atomic orbitals of the separated atoms or ions

of the pertinent system. This approach of linear aombination of

atomia orbitals (LCAO) has been described in the paper of Root­

haan9 who used it in the moleauZar orbital (MO) method.

The matrices Fa and FB are dependent on the density matri­

ces P and Q as can be seen from equation (2.14a-e), which in

their turn depend on the column vectors c~ and c~. To obtain a -1 -1

self-consistent solution for the P and Q matrix, we have to use

an iteration procedure based on the following steps:

(1) assume a set of coefficients c~, c~, satisfying the neces--1 -1

sary orthonormality conditions (eq. 2.5}, and compute P and

Q: (2) calculate the matrices J, Ka, KB according to equation

(2.14a-e) and then Fa and FB: (3} solve simultaneously the equations (2.14} for the electrons

with a- and S-spin:l 7

determine from the eigenvalues a new set of eigenvectors c~ -1

(4}

and c~,1 7 and recompute P and Q: -1

(5} repeat step 2 to step 4 until the computed and assumed den-

sity-matrices P and Q agree within a certain limit.

A possible ariterion for self-consistency is: UHF (1) the change in E (evaluated by equation (2.16) or (2.16a))

24

for two iteration-cycles in succession must be below a . 1 -5 t given lim1t, e.g. 0 a.u.;

(2) the change in each element of P and Q for two iteration cy­

cles in succession must be below a given limit, e.g. 10-4 •

2.3 PROJECTED-UNRESTRICTED HARTREE-FOCK SCHEME

The single Slater determinant wave function in the un­

restricted Hartree-Fock method does not usually represent a pure

spin state, i.e. it is no eigenfunction of the total spin-opera­

tor s2 • Here, we can represent the total wave function by a

linear combination of pure spin states:

"'UHF = ~ C 'I' ' L s '+m s '+m m=O

(2.17)

If we assume p > q the lowest spin component s = s' = ~(p-q) and the highest value of the spin s = s' + q. It has been shown

by Amos and Hall 13 that the coefficients C '+ decreases rapidly s m at an increasing value of m. For obtaining a pure spin state

with s = s•, the components with a spins> s' must be removed.

To construct this spin eigenfunction of muLtiptiaity (2s'+1) the

following spin projeation operator 18 can be applied:

rr {s2-k(k+1) }

kfs' s 1 {s 1+1)-k{k+1) (2 .18)

The average expectation value of the total spin-operator s2 de­

noted by <S2

>sd becomes after spin projection:

2 <S >PUHF

f'l''*s 2'1'' dT

f'¥'*'¥ dT (2 .19 )

in which'¥' = os, '¥UHF. (2.19a)

The reduction in equation (2.19) is a consequence of the fact

that Os' commutes with s2 ; moreover, Os, is idempotent, ~.e.

tThroughout this thesis the atomic units (a.u.) will be used for energy and length: I a.u. energy= 27.2107 eV

1 a.u, length = 0.529167 R.

25

2 0

8, = 0

8,. However, this spin-projection technique is so far not

tractable for a larqe system and we shall therefore only inves­tiqate the effect of sinqle annihilationl9t 20 • 21 of the compo­nent with spin multiplicity (2s'+3).The single annihilator As'+J becomes then:

s 2-(s'+l)(s'+2) As'+l = -2(s 1+l) (2.20)

and will remove the component with spin s = s' + 1 from the to­tal wave function wUHF.

The expressions for several quantities before and after sinqle annihilation are qiven in chapter 3.

2.4 SYMMETRY ORBITALS

The solution of the pseudo-eiqenvalue problem (eq. 2.14) may entail difficulties for larqe matrices Fa and F 6 with re­spect to the necessary computation time and the capacity of the

hiqh-speed memory of the computer. Moreover, the computation time of the iteration process will increase as a consequence of the qrowinq number of integrals (eqs. 2.14a-e) which must be

multiplied in each iteration-cycle by the elements of the densi­

ty matrix. Now, we will try to reduce the Hartree-Fock matrix F into a num­

ber of independent "blocks". This subject can be investigated with the help of group theory.22-2s

The system (complex, molecule) has a certain symmetry which can be indicated by its accessory point group. From the set of basis functions xt we can construct linear combinations that transform according to the irreducible representations of the point group of the system in question. This classification is performed with projection operators.25 The linear combinations are called "symmetry orbitaltJ", together they form a symmetry­adapted basis set.

Use of this set of wave functions will split the original matrix Fa and FS into a number of smaller independent matrices(blocks).

The size of each matrix is determined by the number of new basis functions which has be.en classified in the relatinq irreducible representation.

26

We will illustrate this concept for a pure octahedral com­

plex MX:- (symmetry group Oh) with the atomic orbitals: d, s, p as basis functions for the central metal-ion M and likewise 2s, 2p for the ligands X. In this case the size of the Fa and FB­matrix will be 33 x 33. The classification of the basis func­tions according to the irreducible representations of Oh can be seen in table 2.1.

Table 2.1 Symmetry orbitals for a complex with symmetry group Oh

•

irreducible representation

e g

metal orbitals

6

d 2 2 X -y

d 2 z

d xz d yz d xy

combinations of ligand orbitals•

61 + s2 + 63 + 64 + 65 + s6

-xt - Y2 - z3 + x4 + Y5 + z6 /3( s 1 - 62 + 64 - 65 )

/3(-xt + Y2 + x4 - Y5 ) -s 1 - s 2 +263 - 64 - 65 +2s6 xi + Y2 -2z3 - x4 - Y5 +2z6

-zl + x3 + z4 - x6

z2 - YJ - z5 + Y6

- z3 - z6 zl + z2 + z4 + z5

Yt + x2 - Y4 - xs

Yt - Y3 + Y4 - y6 - x2 + x3 - xs + x6

-zl + z2 - z4 + z5

si,xi,yi and zi stand for 2s,2px,2py and 2pz respectively on centre i.

27

Here, the coordinate axes on the different centres of the ions

have all been chosen parallel to the main axis on the central metal-ion (see fig. 2.1).

z

Ys

Fig. 2.1 The choice of the axes for an octahedral complex (metal on position 0, ligands on positions 1-6)

In closed-shell systems and open-shell systems treated by the method of Roothaan,9' 10 the Hartree-Fock operator F(1) has the same symmetry as the total Hamiltonian H of that system. In the spin- and symmetry-unrestricted HF method this needs not necessarily be the case, however. For example, the ground state of a d 1-system (octahedral complex with oh-symmetry) has 2T28 -symmetry, whereas the Hartree-Fock operator of a component of 2T28 will have 2B28-symmetry (D4h) in the UHF method. The classification of the basis functions according to the irre­ducible representations of o4h is shown in table 2.2.

The time required to solve the eigenvalue problems of all

the small matrices is less than that for the 33 x 33 matrix. However, the use of symmetry orbitals in the computer programme is a restriction of the general character of the programme and

28

Table 2.2 Symmetry orbitals for a complex with symmetry group D4h

irreducible metal combinations of representation orbitals ligand orbitals

alg dz2' s sl + 52 + 54 + 55 -x, - Yz + x4 + Ys

53 + 56 - z3 + z6

big d 2 2 sl - 52 + 54 - 55 X -y -x, + Yz + x4 - Ys

a2g Yt - x2 - y4 + xs

a2u Pz 53 - 56 - z3 - z6

z, + z2 + z4 + zs

b2g d Yt + x2 - y4 - x5 xy b2u zl - z2 + z4 - z5 e d zl - z4 g xz

x3 - x6 d yz z2 - zs

y3 - y6 e Px sl - s4 u

-xl - x4

x2 + xs

x3 + x6

Py 52 - ss

- Y2 - Ys

Yt + y4

y3 + y6

is therefore avoided in our problem •. In our computer programme 2 6

we need only the following input data: (1) the coordinates of the nuclei of the ions in the complex; (2)

(3) (4) (~) (6)

the nuclear charge of each ion; the basis functions (Slater-type orbitals) for each ion;

the basls functions that can be fixed ("core" orbitals); the electronic donfiguration;

the \start vectors c~ and c~. -J. -l.

29

The part of the computer programme in which the atomic-orbital integrals (see chapter 4) are evaluated has been composed in such a way that identical integrals occurring in the problem are computed only once.

2. 5 SLATER-TYPE ORBITALS

The spatial part (denoted in spherical coordinates) of a basis function (atomic orbital) x is defined by:

(2.21)

in which Rnl (r) : the normalised radia"t part, i.e. .. 2 2

of r Rnl (r) dr = 17

Y lm ( e '4>) : the normalised angu"tar part7 n,l,m: the quantum numbers of the atomic orbital; r,e,!j>: the spherical coordinates.

The radial part Rnl (r) may be approximated by (a) a Slater-type function or (b) a Gaussian-type function. In our calculations we shall use a linear combination of Slater-type functions:

<;k1" 0

Several papers reveal that even a small number of terms will facilitate a good approximation of the Hartree-Fock atomic or­bitals. With the Gaussian-type function we would require more terms to obtain the same result.

Y1m(e,4>) are the normalised aomp"tex spherical harmonics defined by:

* The indices n and 1 in Ckl,n and tkl will henceforth be avoid­ed in the expressions.

30

P1 (cose) ~ ($) m m {2.23)

and ~m ( ljl) (2.24)

The normalised associated Legendre functions P1m(cose) are de­

fined by:

1 [21 + 1 ~] l . m r d ] 1 +m 2 . 1 P1m{cose) = 2111

- 2- O+iii}T (-sine) l""ii'Cci'S"6 (cos e-1) {2.25)

where -1 ~ m 5 1.

The normalised associated Legendre functions are related to the

unnormalised ones by:

_ [21+1 (1-m)l ]l plm {x) - -2- O+m)l plm (x) (2 .26)

If m = O, P10 (x) = P1 (x) becomes identical with the ordinary Legendre polynomials. For these conventions the following identities are valid:

Pl (x) = (-1)m (1-m)l Plm (x) (2.27a) ,-m (l+m)l

Pl (x) ,-m (-1)m plm (x) (2.27b)

• (-1)m Yl (e,ljl) (2.27c) Ylm(a,<Pl = ,-m

Throughout the present thesis these conventions of Rose 27 t for spherical harmonics are used.

The PeaZ angular functions s 1m(e,ljl) can be obtained by lin­

ear combination of the complex spherical harmonics Y1mce,ljl):

{Yl,-lml m

slm (a,+) = N + (-1) ~~:m.Yl,lml} m (2.28 )

where K 1 m for m ~ o, K m -1 for m < o, N i/12 for m < o, m (2.28a)

Nm % for m= o, N = 1/12 m for m > o.

t Note, the conventions of Rose 2 7 are different from those of Margenau and Murphy, 8 viz. Ylm (Margenau) • (-l)m Ylm (Rose),

31

In this thesis the ket-notation of Dirac 7 IX> will only be used

for complex orbitals, while for real orbitals IX) will be used. This distinction has. been made to prevent misunderstandings in the expressions of integrals and accessory numerical data.

Those real orbitals that are essential for a discussion on transition-metal compounds, are assembled in table 2.3 which shows also their relation with the spherical harmonics.

Table 2.3

Ins)

lnp ) X

lnp ) z

lnp ) y

Essential orbitals for transition-metal complexes

- Rns8 oo = Rns 1oo

- Rnp8 11 Rnp r!(Y1-1

- Rnp8 JO = R np

- R Sl I = np - R np

y10

I nd 2 2) -X -y Rnd8 22 = Rnd

lnd ) xz

I ndz 2)

lnd ) yz

lnd ) xy

REFERENCES

1, Hartree, D.R., The Calculation of Atomic Structures, John Wiley and Sons, Inc., New York and Chapman and Hall, Ltd., London (1957).

2. Pople, J.A,, and Nesbet, R.K., J.Chem,Phys. 22, 57JL (1954). 3. LISwdin, P.-o., Ann.Acad.Reg,Sci.Upsalien. 2,127 (1958), 4. Nesbet, R.K., Revs,Modern Phys. 33, 28 (1961). ~.Kaplan, T.A., and Kleiner, W.H.,-phys.Rev. 156, l (1967). 6. Slater, J.C., Quantum Theory of Atomic StruCtUre, McGraw-

Hill, New York, 1960, Vol,I, ~hapter 9 and appendix 16; Vol,II, chapter 17 and bibliography.

7. Dirac, P.A.M., Quantum Mechanics, Oxford (1947), B. Margenau, H., Murphy, G.M., The Mathematics of Physics and

Chemistry, D. Van Nostrand Company, Princeton, New Yersey (1956).

9. Roothaan, C,C,J,, Revs.Modern Phys. 23, 69 (1951). 10, Roothaan, C,C,J,, Revs.Modern Phys. 32, 179 (1960). 11. L6wdin, P.-o., Phys .Rev • .21.. 1509 (1955).

12. L6wdin, P.-o., Revs.Modern Phys. 32, 328 (1960). 13. Amos, A.T., Hall, G.G., Proc.Roy.SO'c. (London) 263A, 483

(1961). 14. Wilkinson, J,H., The Algebraic Eigenvalue Problem, Claren­

don Press, Oxford (1965), chapter 5. IS. Wilkinson, J,H., Numerische Mathematik~. 354 (1962), 16, Wilkinson, J,H., Rounding Errors in A1gebraic Processes,

Notes on Applied Science, No. 32, H.M.s.o., London (1963), chapter 3.

17. RC-Informatie no's 11, 13 (1967), Technische Hogeschool Eindhoven (unpublished),

18. L6wdin, P.-o., Advanc.Chem.Phys. 2, 207 (1959). 19, Amos, A.T., Mol.Phys. 5, 91 (1962). 20. Amos, A,T., Snyder, L.C., J.Chem.Phy.s. 41, 1773 (1964). 21. Sutcliffe, B.T., J,Chem,Phys. 39, 3322 (1963), 22. Cotton, F.A., Chemical Applications of Group Theory, John

Wiley and Sons, Inc., New York, London, Sydney (1966). 23. Heine, V., Group Theory in Quantum Mechanics, Pergamon

Press, Oxford, London, New York and Paris (1960), 24. Hamermesh, M., Group Theory and its Application to Physical

Problems, Addison-Wesley Publishing Company, Inc., Reading, Mass., USA; London, England (1962),

25. Tinkham, M., Group Theory and Quantum Mechanics, McGraw­Hill Book Company, Inc., New York (1964).

26. DeLaat, F.L,M.A,H., Complete Set of Computer programmes for Unrestricted Hartree-Fock Calculations (ALGOL 60), Technische Bogeschool Eindhoven (1968), (unpublished).

27. Rose, M.E,, Elementary Theory of Angular Momemtum, John Wiley and Sons, Inc., New York (1957), Appendix III.

33

3 SINGLE ANNIBILADON FOR A SINGLE DETERMINANT WAVE-FUNcriON

This chapter deals with the fact that the single determi­nant wave function used in the UHF method will in general not be a pure spin-state. To get an idea about this deviation some spin P§Operties such as the average expectation value of the total S spin-operator, and the spin density are considered before and after spin pro~ection (single annihilation). Formulae will be deduced for <S >, the charge-density and spin-density function.

3.1 AVERAGE EXPECTATION VALUE OF THE s2-0PERATOR

3.1.1 <Sa> before single annihilation

Prior to the evaluation of the average expectation va1ue of the total s2 spin-operator after spin-projection, we shall de­duce the expression of that quantity for the unrestricted HF wave function. The average expectation value of the s2-operator, denoted as

a <S > 1 is defined as:

where Y = ' (1,2, ••• , N) is the total wave function of a N-elec­tron system (cf. eq. 2.2).

The s2 -operator can be written as:

N N ls .• s.+2 }:s .• s.

i•l l. l. i<j l. J (3. 2)

The one-electron part in equation (3.1) becomes after applica­tion of the orthonormality of the set of functions {~i}:

34

N <'PI I s •• s.IT> =

i•l 1 1 (3 .3)

For the same reason the two-electron part yields:

N 2<'1'1 L s .. s.l'l'>

i<j 1 J

- <Tjl. (1)Tjl. (2) ls 1.s 21•· (2}!p. (1)>] (3.4) 1 J 1 J

Substitution of the expression of s 1.s2 :

(3.5)

(3.6)

(for the shift operators: s~, s;, s;, and s2, see ref. 1) in equation (3.4) yields:

N N · 2<VI L s •• s.IV> =\I {{2a(m ,m )-1]

i<j 1 J i<j si sj

z . Hence the complete expression for <S > is:

If there are in the N-electron system p a-spin electrons and q electrons with ~-spin (assume p > q), the expression (3.8) can be reduced to:

(3.9)

Substitution of the equations (2.13) and (2.15) into equation (3.9) yields for the average value of the total s2 spin-operator for a single determinant wave function:

35

3 <8 >sd

2 ~(p+q) + \(p-q) - tr(PSQS) (3.10)

where tr is the trace of the matrix, and S the overlap matrix of

the basis functions Xt•

For the special case of the restricted HF method we have ~~=~~' ~ ~

so that equation (3.10) can be simplified:

(3.11)

in which s' stands for the spin of theN-electron system: s' = 'J(p-q).

3.1.2 <83 > after single annihilation

The average value for <83

> after single annihilation As'+! (cf. section 2.3) is:

<8 2> = <Y'I82 I~'> <~I I 'I''>

(3 .12)

We want to annihilate the component with a spin s = s' + 1 and assume that the intervention of spin components with s > s' + 1

may be neglected because of their much higher energy. A '+I . 2 s

then be taken as 1.-dempotent, i.e. As'+l = As'+l. Moreover, commutes with As'+l so that equation (3.12) reduces to:

<'I'I821As'+l'l'>

<'flAs'+l'l'> (3.13)

We denoted <82> under these circumstances by <82> • Substitu-asa

tion of the expression (2.20) for As'+l in equation- (3.13) yields: 2 ' 3

<82> 1 4 (s'+1) (s'+2)<82>

8d} =X {-<S >sd + as a

= <82> -1 {<S4>sd + <S2>!d} sd X (3 .14 )

where X ... -<82> sd

+ (s'+1) (s'+2) (3.14a)

36

The expression

<S 2>sd (see ref. get: 5

for <S4>sd can be derived on the analogy of 4) and after substitution in equation (3.14) we

(3 .15 )

in which L = tr(PSQS). (3.15a)

Avoidance of the assumption A!'+l = As'+l in the beginning of this paragraph would have given a much more complicated ex­pression for <S2

>. Amos and Snyderq have worked out this expres-2 sion and denoted the corresponding expectation value for the S -

operator by <S 2> • In the calculations figuring in chapter 6 of a a

this thesis we shall use formula (3.15).

3.2 TOTAL ELECTRONIC ENERGY

The total wave function obtained by the UHF method will have a total gies of the

electronic energy which is a mixture of the ener­components with a spins= s', s'+1, s'+2,... The

energy of the components with s > s' is much higher than that of~ the component with a spins= s'. Thus, single annihilation will decrease the total electronic energy obtained by the UHF method.

Again, using the assumption A:'+t = As'+l we arrive at an ex­pression for the electronic energy after single annihilation:

<E> as a

<'l'jH!As'+I'!'> = <'l'jAs'+t'!'> (3 ,16)

This expression is worked out in the paper of Amos and Hall 3 for the extra assumption that the set of basis functions {xi} is orthonormal.

3.3 CHARGE-DENSITY AND SPIN-DENSITY FUNCTIONS

The aha~ge-denaity and spin-density functions denoted by q(~) and p(~) respectively are defined by: 4

37

N q <;:> <'I' I I o <;:i ,;:>I'!'>

i=l (3.17a)

N p(;:) = <'1'1 J 2Szio(;:i,;:) 1'1'>

1"'1 (3.17b)

where ;: stands for the spatial coordinates. The equations (3.17a) and (3.17b) can be reduced by using the

orthonormality of the set {1/J.}, and equations (2.13) and (2.15): 1

q (;:) L (P+Q) tu x:<;:>xu<;:> (3 .18a) t ,u·

p(;:) I (P-Q) • (;:) (3 .18b) = X (;:)X t ,u· tu t u

The corresponding expressions after single annihilation become (the derivationt of these equations can be found in refs. 5,6):

q(;:) I (P'+Q') tu • (3.19a) xt<;:>xu(!:) t,u

p (~) L • (3.19b) (P'-Q')tu xt(;:)xu(;:) t,u

with P' p - ~{PSQSP-,(PSQ+QSP)} (3. 20a)

Q' = Q - ~{QSPSQ-,(QSP+PSQ)} (3.20b)

In chapter 6 we shall give graphs of these functions q(;:) and p(;:) for the TiF~- complex. We shall also calculate the total

spin-density ps(;:F) at the F-site and from this the fraotiona~ density fs which is a measure for the isotropic hyperfine para­meter.7'8

t Here also, the assumption A!'+l As '+I is used.

38

REFERENCES

I, Griffith, J.s., The Theory of Transition-Metal Ions, Cam­bridge Univ,Press, London and New York (1961}, p. 11.

2. L8wdin, P.-o., Phys.Rev. 97, 1509 (1955), 3. Amos, A.T., and Hall, G.G~ Proc.Roy,Soc. (London)~.

483 (1961}. 4. Amos, A.T., and Snyder, L.c., J.Chem.Phys. 41, 1773 (1964). 5, Sutcliffe, B.T., J.Chem.Phys. 39, 3322 (1963). 6, Amos, A.T., Mol.Phys. 5, 91 (1962). 7. Freeman, A.J., and Watson, R.E., Phys.Rev.Letters 6, 343

(1961). -8. Watson, R.E., and Freeman, A.J., Phys.Rev. 134A, 1526 (1964),

39

4 MOLECULAR INTEGRALS WITH SLATER· TYPE FUNCTIONS

This chapter starts with the discussion of some basic con­cepts and gives the complete formulae (in a few cases together with their derivation) for the one- and two-centre one- and two­electron integrals (composed out of Slater-type basis functions) as well as for the three-centre nuclear-attraction integrals. For the two-centre exchange and the three-centre nuclear-attrac­tion integrals a modified zeta-function expansion has been ap­plied, The three- and four-centre two-electron integrals are ap­proximated by several methods, The last two sections deal with the relation between integrals composed of complex functions and those composed of real func­tions as well as the integral expression after an arbitrary ro­tation of the coordinate axes.

4. 1 GENERAL CONCEPTS

The moteauZaP integPaZs in the unrestricted Hartree-Fock method (see section 2.2) will be composed in our case out of Stater-type basis functions, i.e. we use a multi-centre basis set. Corresponding to the number of different centres in the in­tegral we shall divide the set of integrals in one-, two-, three- and four-centre integrals. A subdivision can be made as to the number of electrons occurring in the integral (one- and two-electron integrals). A lated to the unrestricted table 4.1.

survey of the molecular integrals re­Hartree-Fock method can be found in

The following abbreviations have been chosen for the molecular integrals:

<AjMjB > - f x~A)*(1) M X ~B) (1} dv 1 ( 4.1 a) ~ J

<AIMIA' > - f x~A)*<1> M x~A).(1) dv1 (4 .1b) ~ J

<ABjjCD> Jfx~A)*(1) x~B)(1) -I x<c> (2) xi D) • (2) dv1dv2 (4.1c) - rl2 ~ J k

where the index (A) represents the centre of the orbital x and M i.s a one-electron operator.

40

Table 4.1 Survey of the molecular integrals in the UHF method

number of centres

one:

two:

three:

four:

one-electron integral

<AlA'>

<AI-~ll lA' >

<A 1-l" ~I I A f >

<AjB>

<AI-r;1

1A'>

<Aj-~ll IB >

<AI-r;1

1B >

<AI r~ 1 1B >

two-electron integral

<AA'IIA"A"' >

<AA'II BB' >

<AA'II A"B >

<AB II A'B'>

<AB II CC' >

<AB II A'C >

<AB II CD >

Before deriving analytical expressions for the molecular integrals in table 4.1 we introduce three concepts: {1) the spheroidal coordinates; {2) the V 1 {t,t)-functions, and

n mp (3) the Gaunt coefficients.

4.1.1 The spheroidal coordinates

A aharne distl"ibution O~~B) (A) (B) for an electron is ., l.J - Xi X j described in the Cartesian coordinate system by four independent

parameters, for example xA, yA' zA and the internuclear distance RAB' some other possibility being the description with spherical coordinates. Both methods have the disadvantage of giving very complicated expressions after integration over the entire space. Introduction of the sphel"oidaZ coordinate system (see fig. 4.1)

appears to have several advantages. The spheroidal coordinates t, n and ~ are defined by (cf. ref. 1-3):

( 4. 2a)

41

and conversely:

rB = l..iRAB(E:-n) rA = l..iRAB(F,;+n)

coseA = (1+E;n)/(E;+n) coseB = (1-E:n)/(E:-n) (4.2b)

sinaA ={(E: 2;..1)(1 ... n 2 )l~/(E:+n) sinSB ~{(E: 2 ;..1)~1 ... n2 )}1/(E:-n)

xs

Fig. 4.1 The spheroidal coordinate system

For integration in these coordinates, the volume element dv is

(RAB/2) 3(F,;

2-n 2)dE;dnd' and the integration limits for ' are: 0 to 21r, for n: -1 to 1, and for F,;: 1 to "'• The foci of this coordi­nate system coincide with the nuclei A and B which is very con­

venient in applying numerical integration procedures. The expressions for the Slater-type orbitals (STO's) on the centres A and B in these special coordinates can be found 3 by

substitution of the equations (2.23) and (4.2b) in:

= N. ~

in which Ni =

n.+! (21;i) ~

{ ( 2ni) I } I (B) A similar formula can be derived for xj (E:,n,,).

42

(4. 3)

4.1.2 The V 1 (t,T)-functions n mp

The auxiliary functions V 1 (t,•) appear if an orbital (B) n mp

xi is expanded around another centre A at an internuclear dis-tance RAB' In this expansion the coordinate systems on the cen-· tres A and Bare chosen according to fig. 4,1 and the v 1 (t,T)

n mp functions are defined by (cf. ref. 4-8):

(4.4)

where t and T are the lesser and greater of ~irA and ~iRAB re­spectively. The analytical expressions for the V 1 (t,<)-functions with

n. BmBp arbitrary parameters n1,15 ,m5 and p cAn be obtained by using an expansion in terms of the spherical Bessel functions given by Watson: 9

~ (2p+1) i (t) k (•) P (coseA) p=O P p P

( 4. 5}

in which i (t) and k (<) are spherical Bessel functions of the p p first and second kind, defined by:

i 0 (t) sinh t it (t) cosh t sinh t = t t t2 (4. 6)

-T _, + !, ko<•>

e kJ ( T) = ~ (1 = --T T t

with the following recurrence relations:4

(4. 7)

43

In computing the ip(t)-values by means of equation (4.7),

one has to be careful not to loose relevant figures, because the sign of the terms is alternating. A method avoiding this diffi­culty, makes use of the ratios 1 0 of the ip(t)-values with the

accessory recursion relation and is described by Corbat6. 11

The P1m(cose) functions for various 1 and m ~ O, arising in equation (4.4), can be determined from either the definition

(eq. 2.257 2.26) or the following recurrence relations for

Legendre functions:

Pl+t.m(x) = {3.!..!.!_} p ( ) {1+m } p ( ) 1-m+l X 1m x - 1-m+l 1-1 m x •

(4. 8)

The recursion formula for m has been avoided, because we do not require it in our deviations.

For the special case ni=lB=mB= 0 the expression for v000P(t,<) follows directly from the equations (4.4) and (4.5):

rA (2p+1) i (t) k (T) p p (4. 9)

The VnOOp(t,<)-functions with n > 0 are now o]:)tained by repeated differentiation of equation (4.5) with respect to ~i and subse­

quent application of the recurrence formulae for the Legendre functions.l 2 Analogous to this we can derive the expressions for

the remaining V 1 (t,<)-functions. n mp The spherical Bessel functions will often be replaced by the Bessel functions of the first and second kind of imaginary argu­

ment and half integral order Ip+l(t) and Kp+l(T) respectively.

k {T) p

(4.10)

The Vnlmp(t,,)-functions for 1 = 0,1 and n ~ 3 have been derived in this way and put together in table 4.2.

44

Table 4.2 Analytical expressions for V 1 (t,t), arising in ni BmBp

the expansion of a Is-, 2s-, 3s- and 2p-orbital on

centre B, around centre A

• rA(2p+1) Ip+l (t) Kp+! (;)

VOOOp = ItT

r A (2p+1) [t IE+l(t) KE_!(T) I +1 (t) K +i(T)]

VIOOp - t~2 E

ItT ,l't;

vzoop = (t2+,2) VOOOp - 2tT(~ VOOO,p•l +~ 2p+3 VOOO,p+l)

v300p (t2+,2)

VIOOp - 2tT{2p~l VIOO,p-1 ~ + p VIOO,p+l)

v21op = l;iRAB VI OOp - c;irA (-..R..... v .....£.:!:.!. VIOO,p+l) 2p-1 IOO,p-1 + 2p+3

v310p = r;;iRAB vzoop - tirA ( p v 2p-l 200,p-l

+ .....£.:!:.!. 2p+3 vzoo,p+I>

V = r(-1-v 1 v ) 311p r;;i A 2p-1 200,p-1 - 2p+3 200,p+l

• The (t,<)-parameters have been avoided for brevity.

The expansion of a Slater-type orbital around an arbitrary

centre A according to equation (4.4) develops into an infinite number of terms. To get an idea about the rate of convergency of this summation, we have studied the expansion method of a 2s and

2p Slater-type orbital ofF- 13 ' 14 at a fixed distance AB for an

increasing number of terms. The graphs for these radial func­tions R(r) are reproduced in the figs. 4.2 and 4.3.

4.1.3 Gaunt coefficients

The Gaunt coefficients will frequently be used in the ex-

45

46

1.8

1.6

1.4

1.2

ID

0.8

06

0.4

02

f 0 A B

2 3 4

r. t. '1"'\

··--·~- 2s-orbital

.. -15

-10

-5

.. =0

5 7 Rlrl r in a.u. __...,.

-02

Fig. 4.2 Expansion of the 2s radial-part of a F around another centre

8

ion

Fig. 4.3 Expansion of the 2p radial-part of a F ion around another centre

-1 pressions for the integrals with the v 12-operator. The notation

for the coefficients is cA(lAmA,lAmA) and their definition: 1 5 16

cA(lAmA;lA_mA.l

This equation (4.11) is not symmetric in (lA,mA) and (lA,mA), which follows after substitution of equation (2.27b):

(4.12)

Some other properties used in the derivation of the integral ex­pressions later on are:

(4.13)

The conditions for cA(lAmA;lAmA) + 0 are given by the triangu­lar rule, viz. A, lA, and lA must equal the sides of a triangle of even perimeter. So A must satisfy the conditions:

(4.14) 11 -1' I< A~ 1 + 1'

A A - A A

following directly from the definition (eq. 4.11) of_ the Gaunt coefficients.

To obtain an analytical expression for the Gaunt co~ffi­

cients, we must modify equation (4.11) by using the identity:

P {x) = (-1)l<lml-m) P I l(x) lm 1, m (4.15)

47

in order to arrive at:.

(4.16)

where x = cosa. The integral part in equation (4.16) has been worked out by Gauntl5• 16 and gives us after substitution in equation (4.16)

A the general analytical expression for the c (lAmA,lAmA)-coeffi-cients.

[ ( 21 + 1 ) ( 21 '+ l ) ( 21 11 +I ) (1" -m") I (1 +m) I ] ~

• 2 ( 1" +m n) I (1-m) I ( 1 I +m' ) I (1 ' -m I ) I •

(-l)t(l"+m"+t)l (1+1 1 -m"-t)l L (1 11 -m"- t ) I ( 1-1 1 +m 11 + t ) I ( 1 1 -m 1 - t ) I t I t

(4.17)

with lmAI ~ lA and lmll 1 lA 1

g = ~ (lA + lA +A); m", m' and m are chosen in such a way, that m" is the

largest member of the triplet lmAI,ImAI, and !mA-mAI; 1", 1' and 1 are the corresponding members of the triplet

lA, lA and A; the sum over t is to be taken over all terms involving non­

negative factorials.

When these coefficients are evaluated straightforwardly with the help of a digital computer round-off errors will occur.

We therefore have to choose a special representation of the fac­torials in each term, so that multiplication and division of the

48

factorials are reduced to addition and subtraction.~ Rotenberg et al,l 7 describe an analogous procedure for the eval­uation of the 3j-symbols which are closely related to the Gaunt coefficients.

In the next sections expressions will be derived for all molecular integrals of table 4.1. In these integrals we have used comptero orbitats. of a linear combination

Here, the radial part generally consists of Slater-type functions equal to equa-

tion (2,22) with ci coefficients:

n.-1 I ~ i ai rA (4.18)t

The expressions for the reat orbitals can be obtained by appli­cation of equation (2.28).

4.2 ONE-CENTRE INTEGRALS

The one-centre one-etectron integrals 1 -1 operator M = , rA , and -;A can easily be

illustrate this for the overZap integral:

<AlA'>= fx~A)*(1) X~A)(1) dv ~ J

<AIMIA'> with the solved and we will

(4.19)

The set of quantum numbers for xiA) and for xjA) are (lA,mA) and (lA,mA) respectively. Substitution of equation (4.18) in equa­tion (4.19) yields:

<AlA'> = 2! ofnP1 (cosaA) P1 , , (coseA) sineA deA. AmA AmA

( 4. 20)

t (A) The index i' in xi' (r,a,,) refers to the complete sum over

index i on the right-hand side.

49

Integrating over ~A and using the property of the normalised associated Legendre functions P1m(cost!), i.e.:

( 4. 21)

we obtain:

(ni +n. )I <AlA'>= o(lA,lA') o(mA,mA') Z: Z: a.a. 1 i J' ~ J n. +n. +

(1;;.+1;;.) ~ J ~ J

( 4. 22)

Analogously, the analytical expr~ssions one-electron integrals can be derived, -\AlA'> (cf. ref. 3):

for the other one-centre using the expression for

The final expressions will then be:

(n.+n.-1)1 ~ n.+n. (4.24)

<AI-\AIA'> = -\o(lA,lA) o(mA,mA_> •

~;7(n.+n.)l

(1;.+1;.) ~ J 1 J

2n.~,;.(n.+n.-l)l • L Z: a.a. [ J 1 J

• • ~ J n.+n.+l 1 J (1,;.+1,;.) 1 J

- J J 1 J + n.+n.

(1;.+1;;.) 1 J 1 J 1 J

{n.(n.-1)-lA'(lA'+t)} + ] 1

n.+n.-1 (1,;.+1,;.) 1 J

~ J

(n .+n. -2)1 ] 1 J

(4. 25)

The one-centre t~o-eZectron integrals <AA'I IA"A"'> with

quantum numbers (lA,mA), (lA_,mA.), (lA,mA), and (lA''mA') respec­tively, are defined by:

50

Condon and Shortleyl6 give an expression for this type of inte­-1 gral using the following expansion for r 12 :

A A+l where u(rAt'rA 2) = r</r> with r< the lesser of rAt and rA2 and r> the greater of rAt and rAZ' By means of the Gaunt coefficients (eq. 4.11), equation (4.26)

can be written in the following form:

<AA'IIA"A"'> =

= o(m -m• m"-m"') A A' A A

Ri' Rj, Rk, and R1 are x!A) x<A) and x<A) The

J , k 1 • the analytical form:

r~ ffR.R.

]. J

( 4. 28)

the normalised radial parts for x~A), ].

integral part can easily be expressed in

N-;\-1 N+;\ + I (N-;\-1 )I (M+p+A)I- L (N+A)~ (M+q-;\-1 )I]

p=O pi (t{)p+ZA+l q=O ql (l+B)q

in which A(i,j) = ~i+~j B(k,l) = r;k+r; 1

M = mA+mA. N = m"+m"'

A A .

(4.29)

51

This integral (4.29) is for a given set of orbitals only a func­tion of the parameter A and therefore indicated in many books by

A R, called Racah parameter (see ref. 16, p.177).

4.3 TWO-CENTRE INTEGRALS

4.3.1 Two-centre one-electron integrals

The two-centre one-electron integrals are calculated by using a spheroidal coordinate system described in section 4.1.1. After substitution of the equation (4.3) for x~A), of its ana­logon for xjB) and occasionally of equation (4.2) into the in­tegrals, and integration over $, we can either choose to work out the thus resulting expressions by one of the following meth­ods: (1) expressing the integral in an analytical form, using a few

auxiliary functions and then evaluating the resulting for­

mular (2) numerical integration over ~ and n with, for example, the

Gaussian quadrature formula. We will illustrate the first method for the overlap integral <x~A)(1) lx~B)(1)> = s .. for arbitrary x~A) and x~B) with quantum

]. . J l.J ]. J numbers (lA,mA) and (1B 1~) respectively.l•l8-20 Any orbital xi can be described with the three quantum numbers n, 1, and m and a special parameter p. Let us write:

xi (n,l,m,p) =

(4.30)

The coefficients f for the orbitals relevant in our calcula­P

tion, have been tabulated in table 4.3

52

Table 4.3 f -coefficients for some orbitals p

1 m fo

s 0

p 0 1

p ±1 1

d 0 3/2 -~ d ±1 13

d ±2 V6

After integration ·over~' and substitution of equation (4.30),

the overlap integral is written in spheroidal coordinates:

. {(21A+t)(215+t)}j R ni+nj+l

SiJ' = o (mA,lll:s) m +l ~ I aibj { ~B} I L f f •

2 A 1 j p p P A PB

RAB with p = --2-- Ct.+t.)

1 J

RAB p't = --2-- (r;.-t;.)

1 J

By defining now the auxiliary functions:

00

A (p) == J E;q q

Br(p<) Jl r == n -1

equation (4.31) becomes:

in which s = n1-lA+pA

t = nj-1B+p5 u = lA-mA-pA V == lB-mB-pB

e-p!;

e-p<n

w = mA+k2+k~+k 5+k 6

d!;

dn

q = s+t-kl-k2 +k 3+k 4+2k 5

r = k1+k2+k3+k~+2k 6

A B

(4.31 )

(4.31a)

(4.32)

(4.33 )

(4.33a)

53

The symbol <:> is the usual binomial coefficient, that is

(~) al bl (a-b)l (4.34)

The computation of the auxiliary functions A (p) and B (p<) can q r be performed with the following recurrence relations:

A (p) q

P< -p< e -e + !_ B (p<) for r even

P< P< r-l .

(4.35)

(4.36)

It should be noted that the recursion formulae (4.36) may give

large round-off errors for P< < 4, in which case the inverse re-.. cursion formulae, starting with a BN(p<) = 0 for N>>rmax' are recommended.

Taking into account these diffi~fties, the method of nu­merical integration appears to be more convenient and is only slightly slower. In this numerical integration method (for example for the over­lap integral) we start with equation (4.31) and transform the

integral in it, to the corresponding Gaussian-Legendre quadra­ture formula, i.e.:

(4.37)

The values for the coefficients A~n) and corresponding roots x~n) are tabulated in the book of Krylov. 21

The integration limits of ~ in equation (4.31) must be trans­

formed for that purpose to -1 and 1, which can be done with the

relation~= (1+8)/(1-St), in which the parameterS determines

54

the relevant interval of the pertinent charge distribution. In practice the variation of the number n from 16 to 24 turned out to give in general a result with 6-9 reliable figures.

The resulting two-centre one-electron integrals, like

<AI-r; 1 IB> and <AI-~aiB> can be treated in the same manner and will not be discussed further in this thesis.

4,3,2 Two-centre two-electron integrals

The CouZomb and hybrid integrals consecutively defined by:

<AA f II BB' >

with quantum numbers (lA,mA}, (lA,mA), (lB,mB), (lB 1 mB) and (lA,mA), (lA,mA), (l.A_,m.A_), (lB,mB), can be treated simultaneous­ly. In both cases we have a one-centre charge distribution !'l ~~A) = x ~A) • (1) x ~A) (1) for electron 1 and when r -

1 21 is expanded

~J ~ J around centre A (analogous to eq. 4.27), we may write (the deri-vation is given here for the Coulomb integral only): 3

where uCA)(2) is a potential function only dependent on the coordinates of electron 2, and !'l~~B)• x~B) (2)x?)*(2).

Substitution of the expansion for l'~i (eq. 4.27) in equation (4.40) gives:

55

n.+n . • rA

2:a. J. {E + +A ({r;.+l;.)r 1 +A + -A-I [(l;.+t;.)rA2)} (4.41)

ni nj :a. J Az' ni nj :a. J

with \l

(4.42)

The Aq(p)-function has been defined in equation (4.32).

TheE (p)-function can be computed for p>SS with the asymp-P p+l

totic approximation EP(p) "'pl/p , and for S<p<SS with there-currence relation:

I -p in which E0 (p) = P(1-e ).

(4.43)

For p<S the inverse of equation (4.43) has been used, by which

we have supposed that EN(p) = 0 for N ~ 16 + 2p. The limits Amin

and Amax' as well as the increment of A are consequences of the triangular condition (eq. 4.14):

Amax = lA+1A7

Amin =the greater of lmA-mAI and jlA-l,\.1 with the restriction that Amin is of the same parity as Amax'

.Aincr 2 •

If all parts in the Coulomb integral (eq. 4.38a) are written in the spheroidal coordinates and a new variable t is introduced, to treat the integral with the Gaussian-Legendre quadrature for­mula, we have after integration over ~ 2 :

56

N N' [ · RAB 3 2 2 (A) ( ) l l wrws{-2-} 13!;r (!;r-11s) U' (!;r,ns) >~' BB (!; 11 )1(4 44 ) r=l s=l kl . r' s:.l •

in which !;r = (1+13)/(1-atr), 0 ~a::: 1: (4.44a) wr,ws Gaussian weight factors: N ,N' = number of integration points:

th th tr,11s = r and s root of the Legendre polynomial of the th th N and N' degree respectively;

u I (A) (!; 11 ) r' s

Cl. = ( 1.; • + 1.; • ) (!; +11 ) 1. J r s (4.45)

The expression for the hybrid integral is of the same form as equation (4.44), but nk~BB) must be replaced by >lk~AB).

57

0 ,(AB)(" .) kl "r'ns

(4.47)

Roothaan2 has given analytical expressions for the group of in­

tegrals involving 1s-, 2s- and 2p-orbitals.

The exchange integrals <ABI IA'B'> cannot be derived analo­gously to the Coulomb integrals, because the potential function U(2) is no longer dependent on the coordinates of centre A only.

We will therefore take an expansion method of orbital x(B)around

centre A using the auxiliary functions V 1 (t,T) of paragraph n mp

4.1.2. The following procedure in working out this problem is analogous to that of the coulomb integrals (cf. refs. 4,7,8). The e~ohange integrals are defined as:

with quantum numbers (lA,mA)' (lB,mB), (lA,mA) and (lB,mB) re­spectively. Using the general form (eq. 4.15) for P 1 (cose),and the auxiliary function V 1 (t,T) for x~B), the mtwo-centre charge distribution for ele~t~~n 1: n~~B) =\~A)*(1)x~B)(1) be-

1J 1 J comes:

58

-n.+l ?;; • J

J

where L(p,m) [ (p-lml)l]! = (2p+1) (p+ lml )I

(4.49 )

(4.49a)

The expression of the two-centre charge distribution for elec­

tron 2 is comparable to that of electron 1. Taking these ex­-1 pressions together with the expansion expression for r12

(eq. 4.27) and by integration over ,

1 and , 2 , followed by substitu­

:\ tion of the c (lm;l'm') coefficients (eq. 4.16}, we arrive at:

<ABIIA'B'> = o(m -m ;m'-m') L I I I L a~b3.akbl • A B A B A=O i j k 1 ~

In the sum over A, p, and q a number of terms will be equal to zero owing to the triangular condition (eq. 4.14). Introducing this into equation (4.50) yields:

59

<AB!!A'B'>

c"<IAmA;pm8) c"<Iimi;qmB) L(p,m8) L(q,mi)

(4.51 )

where

This kind of integral expressions written in a generalised form can be modified as follows:

.... ofof f(rAI) g(rA2) u (r AI ,r A2) drAtdrA2 =

Of "'f(rAI) r AI X

i\+1 drAI of rA2g(rA2)drA2 + rAJ

+ J oog(rAI) r AI i\

>.+I drAI Of rA2f(rA2)drA2 (4.52) 0

rAJ

When, however, f(x) - g(x} the integral (4.52) can be simplified

to:

( 4. 53)

60

The final integral expression is now evaluated by means of nu­merical integration. To this end the interval of rAJ (outer inte­gral) is divided into three parts in such a way that the break

points rAJ = rA2 and rAJ = RAB coincide with the limits of the integration. On each interval a N-point Gauss-Legendre quadra­ture formula can be applied, if the integration limits have been modified to -1 and +1. An appropriate transformation is for in­stance:

(4.54)

in which a = the under limit of each interval, a = (2 + h)/h, h representing the size of the interval.

The value of a is chosen for the outer integral in most cases 0,

RAB and 2RAB• The integration over rA 2 (inner integral) is also carried out with the help of a Gauss-Legendre quadrature formula (N'points). The intervals for this integral are in their turn determined by .the successive integration points of rAJ' e.g. the (N+1)th inte­gration-point of rAl requires the evaluation of the inner inte­gral-part with limits: t~e Nth point and (N+1)th point of rAJ' which is added to the known part of the inner integral with lim­its 0 and the Nth point of rA

1•

The nummerical integration with N = 1'2 and N' = 2 turns out to give 5 significant figures.

4.4 THREE-CENTRE ONE-ELECTRON INTEGRALS

The three-aentre one-et.eatron integral or three-aentre nu­

atear attraction integral is defined as:

(4.55)

and the corresponding coordinate system is illustrated in fig. 4.4. The quantum numbers are indicated by (lA,mA) and (lB,mB). Again, the orbital x~B) (1) is expanded around centre A by using

J the auxiliary function V 1 (t,<). Introducing this into

nj BmBP

61

equation (4.55) and applying the one-aentPe expansion expression -1

for rc {see ref. 22, p.842):

_ 1 ; ~ <~-1~~1>1 ill(<P-illc) rc = ~., ~., u(rA,RAC)PA {coseA)PA (cosec)e (4.56)

A=O IJ=-A (A+IPI>l ll ll

we obtain, after the choice illc 0, integration over .p and sub­

stitution of equation (4.16):

(4.57)

Here also, the triangular condition {eq. 4.14) will reduce the

number of terms in equation (4.57). The final integral expres­

sion then becomes:

-n.+l . Ha.a. 1;. 3

ij 1. J J

62

Fig. 4.4 Coordinate system for the three-centre one-electron integrals

In the integration interval of rA' there are two break points,

viz. rA = RAB and rA = RAC' This fact has to be taken into ac­count by the choice of the integration limits for the several intervals.

The integration is carried out with the Gauss-Legendre quadra­ture formula.

4,5 APPROXIMATION OF THREE-CENTRE AND FOUR-CENTRE TWO-ELECTRON INTEGRALS

The three- and four-centre two-electron integrals (see ta­

ble 4.1) can be solved by expanding all orbitals around one centre, e.g. centre A, using the expansion method described in

section 4.1.2. The final expressions for these integrals are similar to those of the exchange integrals and contain two and three auxiliary functions V 1 (t,T) for the three- and four-

n mp centre cases respectively. Since the expansion of an orbital around another centre has been derived for the case of z-axes pointing to each other, we must introduce some additional rota­

tion terms. However, the computation time for the set of inte­grals occurring in an octahedral complex like TiF~- is prohibi­tive for an exact computation.* * 22•30

This difficulty also exists for the bipolar expansi~n method in which the rj~-operator is expanded around two cen­tres instead of one.

63

The set of integrals mentioned above turns out to be of such importance in the unrestricted Hartree-Fock calculation of a transition-metal complex (see chapter 5) that they cannot be

neglected. We will approximate them as closely as possible. Most of the approximations applied in analogous cases start from the

Mulliken approximation for a charge distribution lxfA>xJB)):

Substitution of this equation in the three- and four-centre two­electron integrals furnishes an idea about the magnitude of the

integral values. A survey of the integrals with their order of magnitude (expressed in terms of the overlap) is given in ta­ble 4.4.

Table 4.4 Order of magnitude of three- and four-centre two-electron integrals

integral

(ABI!cc ) (AB II cc') (ACI!Bc > (Acllsc ') (AB I leo)

proportional to

(AI B)

(A!B) (CjC') (AIC) (BIC ) CAlc> CBIC') (AlB> (C!D )

The integrals (ABI ICC) are in general the largest ones, so that we will discuss these in detail. The Mulliken approximation yields for this type of integral:

CABI!cc) = %<AlB> { CAAIIcc) + (BBI!cc)} (4. 60)

We have compared the values of this approximation with (a) the exact integrals* of the CH4 molecule and (b) a set of exact in-

t The notation lAB) indicates that the orbitals A and B are real in contrast to lAB> where A and B are complex.

• The exact integrals of CH4 were provided by Prof. P. Ros.

64

tegrals including arbitrary 3d-basis functions (a test case of a computer programme of MIT). The results table 4.5; the coordinate system being 4.5b.

have been collected in shown in figs. 4.5a and

Table 4.5 Results of sever$1 approximations for (ABI ICC)

<• 1 • 4 II 1s 1sl

<• 1 o4 II 2s 2ol

<• 1 • 4 ll2p,ap,l

<•. •• ll2p,2p.)

<•1 • 4 II • 2 • 2>

<•, 1a II "• "•' <•1 2s II o4 o41

<•1 2p,. II •4 • 4>

<•1 2p• II •4 • 4 1

(d• 2 • II Py p7 1

(du 8 II p7

p7

l

(dx2_72

•11 Py P7 l

(d• 2 a II p 1 p,l

(du a II P, P0 l

!d 2 2•11 P, p 1 l X •y

(d 2 s II Px Pal

(d:. s II Pa Pxl

(d 2 2sll Px P,l • -y

0.107463

0.089445

0.088131

0.086393

0,079436

0.079436

0.108207

' 0,108207

0.108207

appr~ of eq. 4,61

0.108029

0,099330

0.099330

appr. of eq. 4.62

0,108029

0.099330

0.099330

~(AI B)

0.091650

0.091650

0.091650

0.114603

0.114603

0.114603

0,092071 0,079436 0,108207 0,099330 0,099330 0.091650 0.114603

0.058995 0.053824 0.060250 0,059793 0,059793 0,091650 0.101815

0.030086 0.025079 0.030937 0,030794 0,030815 0.032785 0.040284

0.185977 0.187902 0.192666 0.184034 0,182388 0,258461 0.250875

0,072553 0.102341. 0.068454 0,065387 0,064803 0.140771 0,089136

0.117975 0.102341 0.124707 0.119120 0.118054 0.140771 0.162384

•0,011880 •0,013223 .0.011926 •0.0117U •0,011726 -0.028760 -0.025504

0.003261 o.oooooo 0,003390 0.003347 0.003344 o.oooooo 0.007462

0,021673 0.021554 0,021899 0.021685 0,021678 0.049814 0.050101

-0.012046 -0.013775 .0.011926 -0.012174 •0.012215 -0.028760 -0.025504

0.003490 o.oooooo 0,003390 0.003447 0.003455 0.000000 0.007462

0.022189 0.022103 0.021899 0.022199 0.022231 0.049814 0.050101

·0.011965 -0.013285 -0.011926 -0.011813 ·0.011781 ·0.028760 ·0.025504

0.003404 o.oooooo 0.003390 0.003374 0.003369 0.000000 0,007462

0.021922 0.0216U 0,021899 O.O:i1812 0.021787 0.049814 0.050101

z

b

Fig. 4.5

a) Coordinate system for testing of (AB II CC) consisting of 3d-orbitals

b) Coordinate system of CH 4

65

The Mulliken approximation possesses several disadvantages­in comparison with the asymptotic approximation in which the orbital x~C) is considered a point charge on centre C so that

the integral <ABI Icc) = (A\r~ 1 !Bl. These disadvantages are:

(1) all integrals (ABJ \CC) with an overlap (A\Bl = 0 are zero, which is not always correct;

(2) the Mulliken approximation cannot sufficiently reproduce the situation of the overlap distribution of the orbitals x~A) and x~B) with respect to centre c.

l. J A method with similar disadvantages,although to a lesser extent,

is the Ruedenberg approximation23 but this method presents a lot of practical difficulties,

For the integrals (AB\ Icc) we will use a modification of the asymptotic approximation:

(AB\ICC) 1 [ (AA\Jcc)

::: 1:2 (A I r ~ I B) _ 1 CAire \A)

(BB/ICC) J + I

(Bir~ \B) (4.61)

where all factors on the right-hand side will be evaluated ex­

actly. Another possibility was proposed by Richardson:2 4

(AB\!cc) ::: f~B { (AA\!CC) + (BB\Icc)} (4.62 )

in which C (A\r~ 1

IBl f AB ., I I (4.62a)

(Air~ jA) + (B\r~ !Bl

The results which were obtained by these approximation methods can be found in table 4.5.

The remaining three- and four-centre two-electron integrals will be approximated by:

(AB\\CC') :: l:i(A\B) { (AAj\CC') + (BBjjCC')} (4.63a)

(ACI\BC') :: \(A\Cl (B\Cl, { (AA\\BB) + (AAj\C'C') + + (cci!BB) + (CC\\C'C')} (4.63b)

66

(ABIICD} : l:!CAIB> (CID> { (AAIICC) + (AAjiDD} + + (BBI!cc) + (BB!Ioo)} (4.63c)

For a detailed survey of the computation methods for vari­ous molecular integrals, we will refer to the papers of Magnus­son,25 Preuss, 2 6 Huzinga2 7 and Ellison.z8•29

4.6 RELATION BETWEEN INTEGRALS WITH REAL AND COMPLEX FUNCTIONS

For all one- and two-centre "standard" integrals (x. (1) IMI l.

X. (1)) J

(i.e. the expressions of the integrals have been derived

for a special configuration of the coordinate axes on each cen­-1 tre, see fig. 4.1) with operator M = 1, -~A, and -r 1Y holds:

(x. (1) IMix. (1)) l. J

<x. (1) IMix. (1) l. J

(4.64)

This identity follows after substitution of equation (2.28) into the final expressions of the integrals in question (see sections

4.2 and 4.3).

The general expression for the one- and two-centre two­electron standard integrals can be obtained by substituting

equation (2. 28) in <xi (1) xj (1) II xk (2)x 1 (2)).

After applying the constraints given by the Kronecker delta­function in the relating integral expressions, we arrive at a relation between the integral composed of real functions and the

integral consisting of complex functions. Some interesting examples illustrating these relations are:*

(p P liP P ) = ~ <p1rp.ffiiP1rPn> X y X y

• For the elucidatio~ 2f the symbols see table 2.3. The symbols 6,w,cr,w,5 stand for the quantum number m • 2, I, 0, -1, and -2 respectively.

67

= <d d II P P > = <dOdo II p11p11> xy xy x x

(d 2 2 d 2 211 d d ) x -y x -y xy xy

The relation for the three-centre one-electron integrals can be derived in the same way.

4.7 DESCRIPTION OF A REAL ORBITAL IN A ROTATED COORDINATE SYSTEM

All final expressions of the multi-centre integrals in the previous sections have been given for a special choice of the coordinate axes on the relating centres (standard integrals).

However, in general, the molecular integrals in the calculation do not satisfy these conditions and the axes must be rotated.

The rotation of a coordinate system can easily be described with the Eulerian angles a, e andy (see fig. 4.6). The new axes

68

Fig, 4,6 The Eulerian angles a, a and y (x, y, z are the old axes and x',y',z' the new one)

x', y' and z• will then be obtained after a rotation (a) around the original z-axis, followed by a rotation (B) around the new y-axis, and a rotation (Y) around the new z-axis. The direction of the rotations a,B, and y has been indicated in fig. 4.6. With these conventions the transformation matrix for the rota­tio~ is given by:

(

cosacosBcosy-sinasiny ! = sinacosecosy+cosasiny

-sinBcosy

-cosacosBsiny-sinacosy -sinacosesiny+cosacosy

sinBsiny

cosasinB) sinasinB !'

cosB (4.65)

Substitution of equation (4.65) in the real orbitals yields a relation between the orbitals in the oZd and new coordinate sys­tem. Tabl~ 4.6 shows the result for all p- and d-orbitals.

'~

Table 4.6 Components of a real orbital in a rotated coor­dinate system

• {cosacossoosy - a1n~J•iny) IPx')

+ c:osasine IP.,) - (oosacosasiny + sinacosy) lp

1,)

= .... sinecos-y lpx,)

+coss lp1,)

+ sinesiny IP1

,1 = (sin(,lcosscosy + eosasiny) IPx')

+ sinasina I Pz, I

- (sinacosedny - oosacosy) IPy•l

= (cos2acos 2 acos2 y-cos2(,lsin2 -y-sin2acosasin2y+~c0s2asin2 e)

+ \cos2asin2ecosy

+ V3 cos2asin2e

- \ (cos2asin2ssiny+2sin2adnecosy) - -\ (cos2acos2 ssin2 y+cos2asin2y+2sin2acos scos2y)

= (-cosasin2&cos 2r+sinClsinesin2y+'scosasin2S}

+ (eosaeos2ecosy-sinacosasin.y)

+ VJ oosasin2B {cosacos2Ssiny+sinaeos a cosy)

+ (\cosasin26sin2y+sinas1necos2y l

= j-13 sin2a (2cos 2y-sin2r-'ll

- \13 s1n2ecosy + \(2cos2e-sin2el

+ Vl sin28siny + ;13 sin2ysin2a

• ( -sino.sin2SC06 2: y-COS!lS1nBsin2y+'Js1nusin2 $)

+ {sinaeos2e.cosy+eosucosesiny)

+ %/3 sinasin2B + (-sincxcos2ssiny+cosacosacosy)

+ ( ;sina.sin2 asin2 y-cosasinecos2y)

= (sin2acos2 acos2y-sin2~sin2 r+eos2acosSs1n2y+\sin2o:sin2 a) + (~sin2asin2acosy+cos2•sinasiny I

+ Vl sin2asin2a + (-\sin2asin26siny+cos2osinBcosy)

+ ( -~sin2ocos2 asin2y-~sin2asin2y+cos 2 acosBcos2y-sin 2 acos acos2 y)

69

Any inteqral consisting of orbitals defined in arbitrary

orthonormal coordinate systems on the centres in question, can be transformed with the help of table 4.6 in such a way that the

expressions for the relating standard inteqrals may be applied. However, in this case the data of the full set of pertinent

standard integrals must be available in order to get the numeri­cal value of that integral.

The Eulerian angles a, a and y can easily be found if the

axes in the complex have all been chosen parallel to the main

axes on the central (metal) ion. The a and ~ angle will then be

the local '-angle and a-angle which describe the location of

centre B with respect to centre A. It should be noted that the

angle y can be taken zero in the case of the two-centre inte­

grals. If the axes on centre A as well as on centre B have been rotated

over a and a, the new z'-axes will be parallel. Since the ex­pressions for the standard integrals have been derived in a

coordinate system where the z-axes point to each other, we must still invert the z-axis on centre B.

For the integrals with more than two centres, the angle y

is no longer zero but must be evaluated from the local situation

of centre c with,respect to the centres A and B. The calculation of this angle y requires some geometric manipulation which we shall not discuss further in this thesis.

REFERENCES

I, Mulliken, R.S., Rieke, C.A,, Orloff, D., and Orloff, H., J,Chem.Phys. 17, 1248 (1949).

2. Roothaan, C.C.J., J.Chem.Phys. 19, 1445 (1951). 3. Wahl, A.c., Cade, P.E., and Roothaan, C.C.J., J.Chem.Phys.

41, 2578 (1964). 4. Corbat6, F,J,, and Switendick, A.C., Methods in Computa­

tional Physics 2, p.l55 (B. Alder, S. Fernbach and M. Rotenberg, eds.), Academic Press, New York and London (1963).

5. Barnett, M.P., ibid., p.93. 6, Barnett, M.P., and Coulson, C.A., Phil.Trans.Roy.Soc.

(London) 243, 221 (1951). f,, Harris, F.E., and Michels, H.H., J.Chem.Phys. 43, 8165

(1965).

70

8, Harris, F.E., and Michels, H.H., J.Chem.Phys. ~. 116 (1966).

9. Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge Univ.Press, London and New York (1962), p.366.

10. Miller, J.C.P., British Association for the Advancement of Science, Mathematical Tables, Vol.X, Bessel Functions, Part II, Cambridge University Press, London and New York (1952),p.xvi.

11. Corbato, F.J., J,Chem.Phys. 24, 452 (1956), 12. Whittaker, E.T., and Watson,~.N., A Course of Modern Anal­

ysis, Cambridge Univ.Press, London and New York (1958), p.308, 321.

13, Allen, L.C., J.Chem.Phys. 34, 1156 (1961). 14. Lohr Jr., L.L., and Allen,-r.c., J.Chem.Phys. ~. 2106L

(1963). IS, Gaunt, J.A., Phil.Trans.Roy.Soc. (London) A228, 195 (1929). 16. Condon, E,U., and Shortley, G.H., The Theory of Atomic Spec­

tra, Cambridge Univ.Press, London and New York (1953), p. 17 5.

17. Rotenberg, M., Bivins, R., Metropolis, N., and Wooten Jr., J.K., The 3-j and 6-j Symbols, Technology Press, MIT (1959).

18. Jaff~, H.H., J.Chem.Phys. 21, 258 (1953). 19. Craig, D.P., Maccoll, A., Nyholm, R.S., Orgel, L.E., and

Sutton, L.E., J.Chem.Soc., 354 (1954). 20. Lofthus, A., Mol.Phys. S, 105 (1962). 21. Krylov, V.I., Approximate Calculation of Integrals, The

Macmillan Company, New York (1962) p.337. 22. Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B., Molecu­

lar Theory of Gases and Liquids, John Wiley and Sons, Inc., New York and Chapman and Hall, Ltd., London (1957) p.843.

23. Ruedenberg, K., J.Chem.Phys. 19, J433L (1951). 24. Richardson, J.W., (private coiiiiiiunication). 25. Magnusson, E.A., Rev.Pure and Appl.Chem. 14, 57 (1964). 26. Preuss, H., Integraltafeln zur Quantenchemie, Springer,

Berlin (1956-1961), Vols.I-IV. 27. Huzinaga, S., Suppl.Progress Theor.Phys. 40, 52 (1967). 28, Ellison, F.O,, and Shull, H., J,Chem.Phys:-23, 2348 (1955). 29. Ellison, F.O., J.Chem.Phys. 23, 2358 (1955):-30, Ruedenberg, K., Theoret,chim:Acta (Berl.) l• 359 (1967).

71

S A STUDY OF THE APPROXIMATIONS IN AN UNRESTRICfED HARTREE-FOCK

CALCULATION ON TiF 3-6

This chapter describes the computation scheme of a spin­and symmetry-unrestricted Hartree-Fock calculation. The basis sets for Ti and F used in the calculations on TiF~- are dis­cussed. It is shown that a point-charge approximation for the inner-electrons gives unsatisfactory results. Further, a scheme of several types of approximation for the three- and four-centre two-electron integrals has been set up. Computed values for the total electronic energy E and the crystal-field splitting para­meter IODq obtained with the help of each of these approxima­tions are given.

5.1 SURVEY OF THE COMPUTATION SCHEME

This section deals with the computation scheme for the spin- and symmetry-unrestricted Hartree-Fock calculation with a single-determinant wave function as described in chapter 2. In our calculations, the eigenvectors will have the symmetry of the one-electron Hartree-Fock operator and, in general, this will be different from the point group of the entire system. 1

For systems with paired electrons the spin- and symmetry unre­stricted Hartree-Fock calculation will be identical with the re­stricted one.

The computation scheme (fig. 5.1) used in this thesis, con­sists of the following parts: (1) Input data

72

These data are concercned with the coordinates of the nuclei in the N~electron system, the number of electrons with a­spin and those with B-spin and the set of basis functions. Moreover, the information which of these basis functions may be fixed, i.e. the set of eigenvectors composed of these functions ("core" basis functions) that do not alter throughout the iteration process. We shall indicate all basis functions with the exception of the core functions as "valence" basis functions. Further data are the electronic configuration of the complex and a set of start vectors for the iteration procedure.

Fig. 5.1 Schematical representation of the unrestricted Hartree-Fock molecular-orbital computation •(SCF-criterion and extrapolation procedure)

We shall assume that the coordinate axes on all nuclei are chosen parallel to those on the central (metal) ion (see

fig. 2.1). (2a) Standard integraZs

In this part, all required standard molecular-integrals of section 2.2 (the expressions can be found in chapter 4) are evaluated. The programme for these evaluations is set up in

73

such a way that the integrals appearing more than once will not be computed a second time. In connection with the ex­tent of the high-speed memory (32K) and the storage capaci­ty (drum with a capacity of 524,000 words) of the computer together with the demand for a tractable computation time it has been decided to evaluate exactly the one-electron integrals and the one- and two-centre two-electron inte­grals. The remaining integrals (three- and four-centre two­electron integrals) are approximated in some way (see sec­tion 4.5). The method of approximation as well as the effect on both the total electronic energy and the crystal-field splitting parameter 10Dq will be discussed in detail in section 5.4.

(2b) Rotation procedure

Here, the molecular integrals of the pertinent system are expressed in the standard integrals (see section 4.7) and after evaluation the numerical values* are stored on a drum. The sequence in each set of integrals (one-electron and two-electron integrals) may be completely arbitrary. The computed one-electron integrals are put directly in the matrices s and H. In our calculations we assume that the core and valence electrons are completely separated. So the one- and two­centre two-electron integrals can be divided in (a) inter­action integrals between core and valence basis-functions and (b) interaction integrals between all valence basis­functions. Both sets of integrals have been denoted in fig. 5.1 by "core" and "valence" respectively. The "core"-integrals are put in the matrix H. The three- and four-centre two-electron integrals are now approximated from the two- and three-centre one-electron integrals and the one-· and two-centre two-electron inte­grals.

*The indices i,j,k and 1 of the relating integral (the indi­ces refer to the basis functions) together with the numer­ical value of the integral are stored on the drum. We only require out of the one-electron integrals those with i~j and out of the two-electron integrals those with indices i~j, k~l, i~k with j~l for i s k.

74

(3) Iteration procedure The schematical representation of this process can be found in fig. 5.1, the iteration-cycle being indicated by an ar­row. To solve the pseudo-eigenvalue equations the start matrices ca and c6 (or a density matrix P and Q). are re­quired. For this purpose diagonal matrices P and Q belong­ing to a complete ionic model for the complex are frequent­ly used. This assumption is equivalent to the use of an "ionic Hamiltonian". To compute the matrices J, Ka and Kf! and hence Fa, Ff! the integrals are taken successively from the drum and multiplied by the relating elements of P and Q. The pseudo-eigenvalue problem for the a- and a-set are solved simultaneously by the method of Wilkinson.2-s This yields two sets of eigenvalues with corresponding sets of eigenvectors and hence new matrices P' and Q'. The complete iteration-cycle is then continued until the matrices P and Q are converged. The criterion for convergence can be found in paragraph 2.2.2 and section 6.1.

For this general computation scheme a set of computer pro­grammes has been developed6 in ALGOL 60. The iteration-programme in it necessitates a 32K high-speed computer and a drum of about 500,000 words. With the help of these programmes we perform a spin- and symmetry-unrestricted Hartree-Fock calculation on a pure octahedral TiF~- complex (in principle suitable for each other complex).

The purpose of this calculation is to get an insight into the electronic structure of a complex by means of a non-empiri­caZ approach. We shall investigate for this the following ques­tions: (1) what are the errors in the various matrix-elements caused

by a point-charge approximation for the core electrons~ (2) in how far is the computed total electronic energy depen­

dent on the method of approximation for the three- and four-centre two-electron integrals,

(3) to what extent does this question apply to the crystal­field splitting parameter 10Dq7

(4) what is the influence of an incomplete set of basis func­tions for the Ti and F ion on the total electronic energy,

75

. 2 the parameter 10Dq, <S >, the charge-density and spin-den-

sity functions, and the orbital populations;

(5) in literature, several methods are used to evaluate the

crystal-field splitting parameter. Are the results obtained

by these methods meaningful;

(6) is it possible to compute the correct behaviour of the to­

tal energy as a function of the metal-ligand distance for 3-the TiF6 complex in the ground state and first excited

state and hence the crystal-field splitting parameter;

(7) can the factors of the Wolfsberg-Helmholz approximation ap­

plied in semi-empirical calculations be rationalised;

(8} what is the effect of a spin projection on the numerical 2 values of <S >, and on the charge and ·spin density?

The research mentioned above will be described in two parts:

(a) The treatment of the core-electrons and the importance of

the three- and four-centre two-electron integrals in the

calculation will be discussed in this chapter •

(b)

5.2

use

In this investigation we shall use a number of results ob­tained with different basis sets and at different metal­

ligand distances. From these results we will select two computation methods (called M4 and M7) on the basis of

which we shall continue our. investigation (see chapter 6); chapter 6 contains a systematic investigation into the ef­

fect of the basis set, the metal-ligand distance and the spin projection on the total electronic energy, the para­

meter 10Dq, the orbital populations, the expectation value

of the operators s2, and the charge and spin density.

SELECTION OF BASIS SETS FOR Ti AND F

3-In the calculation of the octahedral TiF6 complex we shall the atomic orbitals of the ions Ti3+ and F- as a set of

basis functions forTi and F. For the Tin+_ions Richardson et

al. 7 ' 8 furnish approximated radial parts all consisting of a small number of exponentials. For the F- basis functions there

are various publications, e.g. the functions of Bagus 9 ranging

from a minimal to a larger more accurate basis set, the SCF­functions of Allen and LohrlO•ll and the SCF-functions of Cle­menti.l2tl3

76

For our calculations we have chosen the functions of Richard­

son7'S forTi and those of Lohr11 for F. The characteristic pa­

rameters k '~k' Ck(see eq. 2.22) of these functions can be found in table 5.1. For some of them the function P(r) = r R(r) has

been plotted in fig. 5.2a-c.

Table 5.1 Basis functions of Ti and F

basis functions of Ti11 basis functions of p*

k ~k c•• k k ck c•• k

1s: 1 21.40 +1.000000 1 s: 1 12.1870 +0.117971 2s: 1 21.40 -0.355890 1 8 .18 90 +0.888998

2 8.05 +1.061441 2 14.2960 -0.002390 2p: 2 8.80 +1.000000 3 3.7500 +0.010110 3s: 1 21.40 +0.136929 2 4 .1211 +0.005290

2 8.05 -0.469477 2 2.7178 -0.012170 3 3.64 +1.093641 2 1.6465 +0.001800

3p: 2 8.80 -0.292522 2s: 1 12.1870 +0.051400 3 3.31 +1.041906 1 8.1890 -0.343568

3d: 3 4.55 +0.495554 2 14.2960 +0.016360 3 2.00 +0.633442 3 3.7500 +0.131489

3d': 3 4.55 +0.439079 2 4.1211 +0 .187209 3 1.60 +0.739664 2 2.7178 +0.478507

4s: 1 21.40 -0.040040 2 1.6465 +0.309608 2 8.05 +0.139731 ·2p: 2 5.9696 +0.076300 3 3.64 -0.371042 2 3.0759 +0.475023 4 1.55 +1.056145 2 1.4357 +0.513523

4p: 2 8.80 +0.073550 2 0.64417 +0.079430 3 3.31 -0.286780 3 4.0500 +0,001040 4 1. 31 +1.037212

• For basis functions of Ti, see refs. 8.9; •• ibid. of F , see refs. I I, I 2,

The coefficients are given to six decimal places, but in the calculation we used more accurate coefficients,i.e. thirteen figures.

77

78

12

to

o.e

0.6

0.4

02

f 0 4 5 6 7 e Plrl r in a.u. __....._

-02

-0.4

·0.6

Fig. 5.2a P(r)-function of a 3s- and 3p-orbital of Ti 3+

1.0

-0.4

Fig. 5.2b P(r}-function of a 3d-. 3d'-. 4s- and 4p-orbital of Ti3+

1.0

OB

ll.6

0.4

Q2

t 0 1 P!r) rinau_......

-02

Fig. 5.Zc P(r)-function of a Zs- and Zp-orbital of F

3-The total number of basis functions for the TiF6 complex is 48 with a total number of electrons of 79. The electrons are divided into 40 electrons with a-spin and 39 with s-spin in the case of a doublet state. Since a calculation, in which all basis functions are taken into account, would require too much computer time, we have decided to fix a number of basis functions (ct'. ref. 14,15). This reduc­tion is based on the assumption that the molecular orbitals of the inner electrons hardly change under the influence of the neighbouring ions. For the F-ion we shall fix the 1s-orbital and for the Ti3+ ion either the orbitals 1s,2s,2p or 1s,2s,2p,3s,3p. The notation codes for these types of basis sets can be found in table 5.2.

Table 5.2 Basis sets of TiF~-

'1'1 basis set

type core valence

A 1s,2s;2p 3s,3p,3d B 1s,2s,2p,3s,3p 3d c 1s,2s,2p,3s,3p 3d,4s,4p

F basis set

core valence

1s 2s,2p

1s 2s,2p 1s 2s,2p

number of:

valence electr. bas.fu.

33 57

29 49

33 49

79

The calculation with the set of basis functions denoted by B and c (see table 5.2) will be carried out for two different radial parts of the 3d-orbital (3d and 3d'). If the basis set includes the 3d'-orbital we will indicate this by B• and c•.

The set of basis functions for Ti as well as for F have been evaluated for each particular ion in such a way that they

3-form an orthonormal set. In the TiF6 complex we have not ortho-gonalised the Ti-functions on the F-functions.

5.3 THE TREATMENT OF THE CORE-ELECTRONS

The use of core-basis functions in the calculation implies an approximation that might have more or less influence on its results. The core electrons can be treated in one of the follow­ing ways: (a) point-charge approximation, i.e. the core electrons are re­

presented by a negative point-charge on the nucleus7togeth­er, they form a new effective positive charge;

(b) fixed basis function approximation, i.e. the molecular or­bitals consisting of the core basis functions are kept con­stant during the SCF-iteration process.

The first method will introduce large errors in the one-centre integrals as can be seen by the following comparison. An elec-

3+ tron in the valence basis function 3d,4s,4p of Ti and 2s,2p of F- will be attracted by its own nucleus as well as repelled by the core electrons. If the number of core electrons is n, the nuclear-attraction term for a valence spin-orbital A will, in general, not vanish through the interaction term of the charge distribution A2 with all core electrons in the point-charge approximation both terms will cansel out. The difference ("one-cent:r>e co:r>e e:r>:r>o:r>") in these two approxima­tions is:

(A 1-n:r· ~I I A) + ~ { (AA II A' A' ) - (AA' I I AA' ) } A'

(5.1)

where the summation A' is over all core spin-orbitals. For the numerical data, see table 5.3. The errors are caused by the incomplete shielding of the core electrons and by the quan­tum mechanical exchange integrals.

80

Table 5.3 Error in one-centre part by point-charge approxima­tion of the core

charge nuclear repulsion "one-centre distribution attraction -exchange core error"

Ti: 3d 3d -18.459093 15.982718 -2.476375

4s4s - 7.521778 6.940362 -0.581416 4p4p - 6.105042 5.897859 -0.207183

F 2s2s -2.822245 2.443887 -0.378358

2p2p -2.323681 2.262724 -0.060957

For the two-centre interactions of a valence electron with

the nuclear charge and corresponding core electrons an analogous

comparison can be made. The difference ("two-aentre aore error") is here given by:

<AI-nr;1

1A') + ~ {(AA'IIBB)- (ABIIA'B)J B

where A: a valence spin-orbital;

B: a core spin-orbital.

(5. 2)

These errors as a function of the metal-ligand distance are

shown in fig. 5.3a-d.

If the value of the "two-centre core error" is divided by

the quantity (Air; 1 I A'}, we obtain a measure for a correction on

the nuclear charge oZ which should be used in the point-charge

approximation. In fig. S.4a-d these oZ-values are represented as

a function of the metal-ligand distance. In the same way we have investigated a point-charge approx­

imation for the core-electrons in the interaction of a Ti-core

and a F-core. The difference in this case in the two approxima~ tions described above will be given by:

in which M core spin-orbital of the metal ;

L core spin-orbital of the ligand ;

ZM nuclear-charge of the metal ;

ZL nuclear-charge of the ligand;

R metal-ligand distance.

81

-QOO

-OJ) I

-OJ)2

-0.03

-OD4

-0,05

-OJ)6

r0Jl7

~ -008 c

~-

~ -OJ)9

~

-000

-OJO

-Ol2

~-Ol4

~-Ol6 c

I

-.......... ., ......... ::;~

/~/ ,/ I

/ I I I

i t

i t I

i I

; l I

i I j I

I i r i I i I

t i I ; I

i I I

i I i t i I

I I I I

r1na.u.-3D

a

~

{=Hg;.~'" Ti _,_lldYldWl

-~~~r.\

l5 4D

F 12pcJ2s) E!lpa2pal

12s2s) 12p!<2pl<l

45

-o2~to~--~----~1o.---~l~s~--,,~o----~,b rinau.-

c

-OJ)O

-(10

-oo

-003

-OJ)4

-oos

-006

-uuu

-OJ)

-0.04

-006.

-0.08

-0.10

·Ol2

, .. , ................................ , .. ____ _;;.::~·-

'/

,.,. ........ _...- .,.....---

/ ,/''//

/ //

i / I / I / I / i/ l/

l /i i I i I i i

rina.u---

b

basis set ate>

~:·:;;.:.::;:,.:~:=;::.=,;_-..:.:: ....... ,..

. / / /

/ / I /

/ I . I l I

I I . I l I I 1 { I . I I I

I I I I I I

.j I

rin au----

d

Fig. 5.3 Two-centre core error for charge distributions of Ti 3+ and p- as a function of the metal-ligand distance

82

U5

04

0.3

02

1 ~ 0.1

.E NS! N ...

I I I \ \ \ \ \ \

' \.

{

-13d•Jdol n -. -· 13pc3po)

---l3s3s) .. "t3dX"3diC)

······· ........... __ ',_ uo20 ·:::::;_;;"'···~· -=~10~'-----:3.5;';:------:,'="o---:45.

rma.u.-...-

a

Q5 I I I basis ~tA

I 04 I

I I I fl2po2sl

\ 12p.2p!)

F 12112sl I l2p<2p<l \ \ \ \

02 \ \

\

lQI

\ \

\ ' ' \ ' E \ ' 0

' '.., " ';;: ' i ·-...

~0 ······· ..•... -·-· 2.5 3.0 3.5 45

rma.u.--

c

10

25

20

15

1D

I ~OS

i I I \ \ \

r>asis set B {(:}

~ .'\. \ ' . \. \ \.

\ ' \ \. . \ \ '\ \ ' . ' \ '

\.. ', '\. ' ,.,_ '..._ ....... ..._ ....... _

' --~o---~275--l~0~-~3.5~-~4~0==---:45 rina.u.--

b

10

119 b.tsis stt BtC)

us \ I

(11 I §'2 •• 211. I F 12p0'2r~' I 12112$

Q6 \ 12p<2pltl

\ \

0.5 \ I \

Q4 \ \ \

0.3 \

" \ ' fQ2 ' \ ' \ ' ' :; '\..

,, SO,l ' ' ' N .........

... __ .. "·- ----

llllJ.o 25 10 10 4D 45 rin.au-

d

· s 4 ~z 1 f d'ff d' 'b · f r· 3+ F~g. • u -va ues or ~ erent ~narge ~str~ ut~ons o ~

and F- as a function of the metal-ligand distance

83

For the case of a Ti-core (C-type) and a F-core at a distance

R=3.0 a.u. the difference was only +0.000235 a.u. Orthogonalisa­tion of the cores will have an equally small effect.

5.4 INFLUENCE OF THE THREE-CENTRE AND FOUR-CENTRE TWO-ELECTRON INTEGRALS ON BOTH THE TOTAL ELECTRONIC ENERGY AND THE 10Dq

PARAMETER

We shall now proceed to the discussion of the results for the unrestricted Hartree-Fock calculation on a pure octahedral

3-TiF6 complex. Using the formulae in chapter 4 the following molecular integrals were computed exactly: the one- and two-cen­tre one- and two-electron integrals, and the three-centre one­electron integrals. In the last type of integrals we have used

an effective nuclear charge* equal to the real nuclear charge diminished by the number of core electrons for the ion in ques­tion (point-charge approximation for nucleus plus core-elec­trons). The three- and four-centre two-electron integrals which can be divided in four sets (see section 4.5) have been approximated. To get an idea about the importance of each of these integrals,

we have included them systematically into the calculation ac­cording to the scheme in table 5.4. The abbreviation Sc in this table means an approximation of the integral (ABj jcc) according

to equation (4.62) with a factor f~B proportional to the overlap (AjB).Further, the factor was only dependent on the situation of the centres A,B and C, and can be considered an average valuet for all possible orbitals on the centres A and B. The abbrevia­tion M>10-4 indicates that the integrals have been approximated by the method of Mulliken and only those which were larger than 10-4 have been included into the calculation.

As a starting-point for the iteration we have assumed in all calculations ah ionic Hamiltonian. After each iteration-cy-

z * The same has been done in the integral (Aj- Beff - jB).

rB rA

t The details of this method are not discussed, since it is an intermediate method between MO and M2, and both are well de­fined,

84

cle the total electronic energy has been computed with the new

matrices P' and Q'.

Before evaluating the numerical value of the total electro­

nic energy , we shall discuss the energy expression for a system in the fixed-core approximation into more detail. If we assume

that the basis functions on the same atom are orthonormal and

likewise for the valence basis functions on the core basis func­tions of another centre, the energy expression of equation

(2.16) can be reduced to (the repulsion of the nuclei is added):

(5. 4)

ex (3 in which: P,Q,G ,G

Gcxc,G(3c are only related to the valence-orbitals1 relate to the core-orbitals as well as the

valence-orbitalsJ

i,j a core spin-orbital of the metal or of the ligandr

A,A' metal or ligand centre.

We will denote the first row in the energy expression (5.4)

by Escf' the one-centre part of the second row by Econst and the remaining part by E • After omitting the constant quantity rep Econst the energy expression will be indicated by Evar·

Table 5.4 Survey* of the approximations in the three- and four­centre two-electron integrals

centre integral MO M1 M2

<ABIIcc l M' Sc As

three (ABjjcC')

(ACjjBC l _,.

M M M>to (ACIIBC') M>1o ..

four (ABI!co)

"two-centre core error" P P P

•As asymptotic approximation (eq. 4.61) M Mulliken approximation (eq. 4.63b,c) M' Mulliken approximation (eq, 4.63a)

M3 M4 MS

As As As

- M>to .. -8 M>to -4 M>l08 - M>lO

NP NP NP

NP fixed basis function approximation for core-electrons

M6

As

_ .. M>to

M>to"

M

NP

P point-charge approximation for core-electrons (two-centre) Sc asymptotic approximation with "average" factors f~B'

M7

As

M' _,. M>to

M>1o"

M

NP

85

With the approximations denoted in table 5.4 calculations will be performed for a component of the ground state 2T28 as well as for a component of the first excited state 2E8 • We have

chosen the d and d 2 2 component respectively. To obtain the xy x -y correct component we must take into account that during the iteration the sequence of the molecular-orbital energy levels may alter.If so, we must adapt the distribution of the electrons in such a way that the correct component is obtained. Only then it is possible to arrive at a self-consistent solution for the

component desired. It has been found that the computed matrix P and Q after each iteration-cycle may be used directly as input in the next cycle.

3-The first series of calculations on the TiF 6 complex we

will discuss have been performed with the basis sets B and C at

a metal-ligand distance R=3.8810 a.u. This distance equals the sum of the ionic radii of Ti3+ and F- and is comparable with the

distance calculated with in the study on TiF~- by Bedon.l6 The method on which the integrals were evaluated covers M1 to M4. The computed value of the variable part in the total electronic

energy expression Evar as well as the parameter 10Dq have been shown in table 5.5. The numbers between the brackets behind Evar

and 10Dq refer to the number of iteration-cycles which have been performed. So, the symbol (0) indicates that the energy has been evaluated with the start matrix P and Q (in this case the energy of the pure ionic model), and (1) indicates that the first new matrices P' and Q' have been used in the energy expression, etc. In the calculations we assumed that the SCF-solution is reached if the difference in was less then 10-6 •

matrix elements of Ca 10-4 or less.

two subsequent values of the energy Evar It has been found that the changes in the

and c8 will under these circumstances be

The second series of calculations have been performed with different basis sets (A,B and C) and the approximations of meth­od M4. The metal-ligand distance was in this case 3.4929 a.u. (10% below 3.8810 a.u.). The computed values can be found in table 5.6. It is obvious that from these results the basis set A yields the highest values for the parameter 10Dq. An extension

86

Table 5.5 Influence of the integrals CAAI IBC) on both the vari­able part of the total energy Evar and IODq

R-3.8810 a.u.

MO: E:ar(1) Evar(scf) 10Dq(1) 1 ODq (scf)

M1: Evar( 1) E (scf) var 10Dq(1)

10Dq(scf)

M2: Evar(1) Evar(scf) 1 ODq (1)

10Dq(scf)

M4: Evar(1) E (scf) var 1 ODq (1)

1 ODq (scf)

** M4: Evar( 1) Evar(scf) 10Dq(1)

10Dq (scf)

basis set B

-147.39606 -147.43720

-147.40097 -147.44226

-0.04114 -0.04129

-148.04183 -148.00051

-148.04249 -148.00172 +0.04132

+0.04077

-147.86768 -147~84113 -147.86848 -147.84179

+0.02655 +0.02669

basis set c

-154.12883 -154.24849

-166.60353 -166.52457 -0,11966

+0.07896

-150.42178 -150.36137

-150.45494 -150.39979 +0.06041

+0.05515

-147.60707 -147.57680 -147.62080 -147.58797

+0.03027

+0.03283

-147.90896 -147.88177

-147.92176 -147.89250 +0.02719 +0.02926

-147.89999 -147.87360 -147.91360 -147.88502

+0.02639

+0.02858

• Evar was evaluated by means Evar and IODq are in atomic

** In the approximation method plied.

of equation (5.4). All _values of units. "As" formula (4.62) has been ap-

of this calculation has therefore been set up for the approxima­tions of the methods M3 to M7. The latter results are shown in table 5,7.

87

Table 5.6 Influence of the basis set onE and IODq* var

basis set•• A basis set B basis set c R-3,4929 a,u.

2T 2E 2T 2E 2T 2E method M4 2g II 2g B 2g ,B

Evar(1) -201,56905 -201.49185 -148.40993 -148.37982 -148.59758 -148.55850

Evar (scf) -201.56949 -201.49227 -148,41240 -148.38103 -148.63522 -148,59052

10Dq(1) +0.07120 +0 ,03011

10Dq(scf) +0,07722 +0.03137

• All values .are i'D atomic unite.

••the difference between the value of Evar with basis aet A and

that with basis set B is caused by the kinetic energy and the

nuelear-attraetioo energy of the 3s- and 3p-electrons.

+0.03908

+0 ,04470

Table 5. 7 Influence of the three- and. four-centre two­electron integrals on Evar and IODq*

R•3.4929 a.u.

basis set A M3 M4 M5 M6 M7

2T2g' Evar(1) -201.73717 -201.56905 -201,56828 -201.53517 -201.53553

Evar (2) -201.76961 -201.56947 -201.56871 -201,53722

Evar(scf) -201.56949 -201.53727

2E Evar (1 ) -201.69181 -201.49185 -201.49106 -201,43403 -201.43304 g

Evar( 2 ) -201.73634 -201.49224 -201.49146 -201,43794

Evar(scfl -201,49227 -201.43799

10Dq(1) +0.04536 +0.07720 +0.07722 +0.10114 +0,10249

10Dq(2) +0.03327 +0.07723 +0.07725 +0.09928

10Dq(scf) +0.07722 +0.09928

•All ~aluea are in atomic uoita.

From the results in this chapter we conclude provisionally: (1} the core-electrons may not be treated by a point-charge ap­

proximation1 (2) the calculations with a Mulliken approximation for the in­

tegrals (ABI \CC) leads after some iteration-cycles to a SCF-solution with a value for the total electronic energy which differs too much from the total energy of the sepa­rated ions;

(3) the three- and four-centre two-electron integrals are so important that they must be included in the calculation,

(4) the effect of the iteration process on the value of the total electronic energy and 10Dq is after the first itera-

88

tion-cycle small if all integrals have been included into

the calculation (cf. method M6, M7)r (5) including the 4s- and 4p-basis function of Ti into the ba­

sis set has almost no effect on the values of the total

energy and 10Dq~ (6) including the 3s- and 3p-basis function of Ti into the

basis set has an unprecedented large effect on the 10Dq.

Since the purpose of this chapter was only to obtain an

idea about the relevancy of the three- and four-centre two-elec­tron integrals in the unrestricted Hartree-Fock calculation on

transition-metal complexes, we do not discuss here the influence

of the basis set into more detail. This will be done in chapter

6, where the effect of the basis set at various metal-ligand distances is considered for the approximation methods M4 andM7.

REFERENCES

I. Nesbet, R.K., Revs.Modern Phys, 33, 28 {1961) .• 2. Wilkinson, J.H., The Algebraic Eigenvalue Problem, Clarendon

Press, Oxford (1965), chapter 5. 3. Wilkinson, J.H., Numerische Mathematik 4, 354 (1962). 4. Wilkinson, J.H., Rounding Errors in Algebraic Processes,

Notes on Applied Science, No. 32, H.M.S.O., London (1963), chapter 3.

5. RC-Informatie 11,13 (1967), Technische Hogeschool Eindhoven (unpublished).

6. DeLaat, F,L.M.A.H., Complete Set of Computer Programmes for Unrestricted Hartree-Fock Calculations (ALGOL 60), Technische Hogeschool Eindhoven (1968), (unpublished),

7. Richardson, J.W., Nieuwpoort, W.C., Powell, R.R., and Edgell W.F., J.Chem.Phys. 36, 1057 (1962).

8. Richardson, J.W. Powell,-a.R., and Nieuwpoort, W.C., J.Chem. Phys, 38, 796 (1963).

9. Bagus, P.S.~(private communication), 10. Allen, L.C., J.Chem.Phys. 34, 1156 (1961). II. Lohr Jr., L.L., Allen, L.c:: J.Chem.Phys. 39, 2106L (1963). 12. Clementi, E., IBM Journal 9, 2 (1965). -13. Clementi, E., Tables of Atomic Functions, supplement to the

paper IBM Journal 9, 2 (1965), see ref. 12. 14. Brion, H., Moser, C., and Yamazaki, M., J.Chem.Phys. lQ,

673 (1959). IS. Sutcliffe, B.T., J.Chem.Phys. 39, 3322 (1963), 16. Bedon, H.D., Horner, S.M., and~yree Jr., S.Y., Inorg.Chem •

.2_, 647 (1964).

89

6 SOME COMPUTED QUANTITIES OF TiF ~- FOR VARIOUS BASIS SETS AND AT

VARIOUS METAL-LIGAND DISTANCES

This chapter deals with various intermediate results of a UHF calculation on TirJ-. Computed values are given for the to­tal electronic energy, the lODq-parameter and the orbital popu­lations. The calculations are carried out with basis sets A,B and C, and at various metal-ligand distances (2.5000 to 4.2691 a. u.). The theoretical equilibrium distance is determined by means of a polynomial of the fourth degree for the ground state 2T 2 as well as for the excited state 2Eg• The factors in the Wolfsblrg­Helmholz approximation are calculated and compared to the pro­posed ones: 1.67 for a-bonding and 2.00 for n-bonding. The ex­pectation value <82> and also the charge-density and spin-densi­ty functions are evaluated before and after spin annihilation. Moreover, the radii of the Ti3+ and ,- ions are determined.

6.1 GENERAL INTRODUCTION

In chapter 5 it is pointed out that in an unrestricted 3-Hartree-Fock calculation on the octahedral TiF6

complex, one

must take into account all molecular-integrals to obtain a value

of 10Dq which is in reasonable agreement with the experimental

data.

On the other hand, chapter 6 deals with calculations based on

the set of functions A,B or C and the approximation scheme M4 or

M7 (for these notations, see tables 5.2 and 5.4). At various

metal-ligand distances computations are carried out on the octa-3- 2 hedral TiF6 complex in the ground state T2 and in the first

2 g excited state Eg. Moreover, the effect of the extension of the

radial part of the 3d-orbital is investigated at a metal-ligand

distance R=3.4929 a.u.

The aomp"lete scheme of calcul-ations is shown in table 6.1.

Before giving the final results of each of these calcula­

tions, we will study the basis functions of Ti and F into more

detail. The functions P(r) = rR(r) of the valence orbitals are

illustrated in fig. 5.2a-c. Now, we want to get an idea about

the overlap of valence

figs. 6.1a-c illustrate

90

basis functions on different centres~

this quantity as a function of the in-

Table 6.1+ Survey of the calculations 3-on TiF 6

type of basis set: R in a.u.

s• c• A B c

2.5000 X X

3.0000 X X X

3.2500 X

3.4929 X X X X X

3.7200 X

3.8810 X X

4.2691 X

'~'The distances are determined by: 3,8810 a.u. equals the sum of the ionic radii (cf.ref.l); 3,4929 a.u. and 4.2691 a.u. are 10% lower and 10% higher than 3.8810 a.u.; 3.72 a.u. is the experimental distance in TiF 3 (see re£.2),

ternuclear distance. From these graphs it can be seen that the cr-overlaps of the 3d,3s and 3p-orbitals of Ti 3+ with a 2s or 2p­orbital of F~ are of the same order of magnitude, which is also the case with F--valence orbitals on neighbouring centres. This

may be seen as a first indication that the 3s and 3p-orbitals of Ti must not be included in the core. To obtain a better view on the final results in the next sec­tions, we indicate some intermediate results from the SCF-solu­tion of a UHF calculation with method of approximation M7 using basis set A and at a metal-ligand distance of 3.72 a.u.

We have taken only a few · diagonal-elements from the matri­ces H,P,Q and F, and listed them in table 6.2. This table con-

2 tains the data of the ground state T2 as well as the first ex-2 g B

cited state E. The accessory sets of eigenvalues {E~} and {E.} g ~ ~

together with the coefficients for the metal orbitals in the eigenvectors are reproduced in table 6.3a,b for the ground state and in table 6,4a,b for the first excited state. The one-elec­tron energies of the separated ions can be found in table 6.5a.

The computed values for various parts of the energy expres­

sion (eq. 5.4) are shown in table 6,5b. The total electronic energy data have been evaluated after each iteration-cycle anal-

91

0.30

! s

0.35

0.10

(05

rinaJ.t--

-O'~ove:rtap

_,_ tr·over!ap

a

-d-overlap

35

{2pl2p)

\2s!2pl

f2st2sl

r:sL 25 _LJ~Q--------~35~------~40'

ril"'a.u.-

b

s 0 2~.5--------~30~------~3~£~------~~

rina.u.-

Fig. 6.1

92

c

Overlap as a function of ~he metal-ligand distance (Ti-orbitals: 3s,3p,3d,4s,4p; F-orbitals: 2s,2p)

ogously to equations (2.16) and (2.16a)1 see table 6.6. The dif­

ference in the values of the total energy of both methods de­creases as the number of iteration-cycles increases.

Table 6.2 A few matrix-elements of H, P, Q, Fa and F~*

ZT 2g

(d,.1

component) 21! g

(dx2-y2 component)

i Hu P .. ou F~i F~. Pu 0ii F~i ll'~i 11 11

3d 2 2 -18.9016 0.0347 0.0327 +0.5858 +0.6067 1.0801 0.0318 -0.0751 0,6381 X -y

3d -18.7619 0,0065 0,0055 +0.6603 +0,6915 0,0076 0.0066 +0.6579 0.6887

"' 3d.2 -18.9016 0.0405 0.0354 +0.5292 +0.5684 0.0407 0.0387 +0. 5290 0.5634

3d -18.7619 0.0065 0,0055 +0.6603 +0.6915 0,0076 0.0066 +0 ,6579 0.6887 yz

3d -18.7619 1.0222 o. 0041 -0.0105 +0. 7538 0.0072 0.0067 +0.6952 o. 7146 xy

3px -20.5283 1.0236 1.0236 -1.4380 -1.3118 1.0236 1.0236 -1.4380 -1.3139

3p. -20,5283 1,0236 1,0236 -1.3941 -1.3809 1.0236 1.0236 -1.3949 -1.3821

3py -20.5283 1.0236 1.0236 -1.4380 -1.3118 1,0236 1.0236 -1.4380 -1.3139

3s -21.5660 1,0584 1. 0581 -2.4463 -2.3742 1. 0584 1.0580 -2.4469 -2.3752

x, -15,0573 0,9910 0,9928 -0.1504 -0.1485 1.0309 0,9922 -0.1537 -0.1402

., -14.6867 1.0011 1.0018 -0.0708 -0.0708 0.9999 1,0005 -o ,0643 -0,0634

yl -14.6867 1.0135 1.0031 -0.0735 -o ,0693 1, 0011 1.0014 -0.0641 -o .o630

., -15.8600 1.0024 1.0025 -0.9610 -0.9799 1.0171 1.0020 -o. 9769 -0.9719

XJ -14.6867 1.0019 1.0025 -0.0849 -0.0852 1,0018 1.0024 -o. 0870 -0.0872

z3 -15.0573 0,9883 o. 9923 -0.1655 ·-0.1665 0.9881 0,9915 -0.1678 -0.1685

YJ -14.6867 1.0019 1.0025 -0.0849 -0.0852 1,0018 1,0024 -0.0870 -o .0872

$3 -15.8600 1.0024 1.0027 -0.9966 -0.9972 1.0020 1,0022 -o. 9987 -o. 9992

• All values of a. F0 and F 6 are in atomic units and were eot'lputed with basis set A. method M7 and at R•3. 72 a.u. The values are taken from the SCI-solution.

93

Table 6.3a Molecular-orbital energies • for the electrons with a-spin in the ground state 2T

2g

.irr. a number 3s 3p" 3p. Jpy dx2-y2 d d.2 dyz dxy repr .. 'i of electr. xz

Jb•a +0.8()401 0 +1.()22

6a18

+0. 75036 0 +0.001 +1.020

3e1

+0. 70138 +0. 334 +0 .951 +0 .951 -0.334

2b2g +0.03470 +0.926

1a28 -0.04620

2e8

-0.04970 +0.004 -o. 012 +0. 012 +0.004

4a2u -().06021 +0 .004

1bzu -0.06119

Seu -0.06175 +0.009 +0.001 -0.001 +0.009

4e -0.07285 2 +0.001 +0,028 u -o.028 +0.001

sa18

-0.08367 +0.245 +0.024

leu -0.10411 +0.109 -0.145 -0.145 -o .1o9

1e -0.10743 2 +0 ,078 +0.013 8 -0.013 +0.078

3a2u -0.11664 +0.187

1b2g -0.12482 +0.406

2blg -0.13418 +0 .188

4a18

-0.14739 -0.029 +0.193

2e -0.96968 2 +0.166 +0.044 u +0.044 -0.166

1blg -().97985 +0. 048

Jatg -o. 9&127 +0 .078 +0 .033

2a2u -0.98439 +0 .192

2a18

-0.99674 +0.067 -0.039

1a2u -1.40409 +0. 976

leu -1.44634 2 +0.083 -o. 976 -0.976 -0.083

1a1g -2.44790 -0.994 +0 .001

4 All values are in atomic units and were computed with basis set A, method M7 and. at R""'3.72 a.u.

94

Table 6.3b Molecular-orbital energies * for the electrons with 13-spin in the ground state 2T

2g

1rr. 8 number 3s Jpx Jp. Jpy dx2-y2 d d.2 d d repr. £i of electr. xz yz xy

Jblg +0.82400 0 +1.023

2b2g +0. 79292 +1. 009

6a18

+0. 78712 0 -0.018 +1.022

Je +0.73191 0 +0.167 +0. 994 g -0,994 +0 .167

1a28

-0,04169

2e8

-0.04979 2 -0.006 +0.011 +0.011 +0.006

Se -0.05808 +0.013 +0.003 u +0.003 -0.013

4a2u -0.06034 +0.005

1b2u -0.06125

4e -0.07267 +0.000 +0.034 u +0,034 -o.ooo

sa18

-o. 086oa +0.244 +0.028

1b2g -0.09679 +0.064

3eu -0.10338 -0.110 +0.153 -0.153 -0.110

1e -0.10710 +0.073 +0 ,007 g -0.007 +0,073

3a2u -0.11759 +0 ,188

2blg -0.13104 +0.174

4a18

-0,14476 -o. 039 +0 .178

2eu -o. 96557 +0, 211 +0.053 +0.053 -0.211

1blg -o. 97880 +0.047

Jalg -o .98034 +0 .078 +0 .033

2azu -0.98470 +0.197

2a1g -o. 99707 +0.070 -o .036

1e., -1.32396 +0. 209 +0. 946 -o. 946 +0.209

1a2u -1.39117 +0.974

1a18

-2.37704 +0.993 +0 .019

•All values at>e in atomic units and were coutputed with basis set A, method M7 and at R•3.72 a.u~

95

Table 6.4a Molecular-orbital energies • for the electrons with a-spin in the first excited state 2E

g

irr,. a number 3s Jp, 3p. 3py dx2-y2 dxo dz2 dyz d repr .. .i of electr. xy

6a111

+0. 75269 0 +0.001 +1. 019

2b2g +0. 73926 0 +1.007

lea +0. 70117 0 +0 .031 -1.007 +1,007 +0,031

3blg +0.12853 +0. 906

1a28 -0.03568

2e8

-0.04591 -a. 001 +0,023 +0 .023 +0,001

Seu -0.05420 2 +0.004 -0.000 +0.000 +0.004

1b2u -0.05439

4a2u -0.05499 +0.012

4e., -0.07567 -0.004 +0.021 -0.021 -0.004

5a18

-0.08125 +0. 245 +0 ,023

1b2g -0.09450 +0.085

Je., -0.10575 2 +0.091 +0.161 +0.161 -0.081

1e11

-0.10763 2 +0.009 +0.083 +0.083 -0.009

3a2u -0.11857 +0.198

4a 18 -0.15181 +0.030 -0.193

2blg -0.19514 +0.503

2e., -0.96437 2 +0,093 -0.139 +0.139 +0.093

3a 18 -o .97591 +0.092 +0 .033

1blg -o. 98359 +0.075

2a2u -o. 99609 +0,193

2a 18 -o. 99961 -o ,062 +0.043

1a2u -1.40499 +0.975

1e., -1.44565 +0.897 +0. 398 +0,398 -o. 997

1a 18 -2.44847 +0.994 +0.001

*All valuea &'t'e in atomic units and were computed with basis set A, method H7 and at ll•3.72 A.u.

96

Table 6.4b Molecular-orbital energies * for the electrons with S-sp in in the first excited state 2E

g

irr. .~ number 3s 3px 3p. 3py d 2 2 dxz d.2 d d repr. 1 of electr. X -y yz xy

3blg +0. 86001 0 +1.024

6a18

+0. 79000 0 -o .020 +1.020

2b2g +0.75842 +1.008

3e +0.73140 +0.047 -1.007 g +1.007 +0.047

1a28

-o. 03453

2e -0.04548 +0.001 -0.024 g -0.024 -0.001

5e -0.05183 +0.015 +0.001 u -0.001 +0.015

1b2u -0.05333

4a2u -0.05431 +0.013

4e -0.07277 +0.004 -0.063 u -0.063 -0.004

5a18

-0.07765 +0.243 +0.044

1b2g -0.09324 +0.082

3e -0.09665 -0.036 +0.179 u +0.179 +0.036

1e -0.10695 +0.005 +0.078 g +0.078 -0.005

3a2u -0.11921 +0.188

2blg -0.12384 +0.171

4a 18 -0.14788 +0.049 -0.184

2eu -0.95724 +0 .207 +0.057 +0 .057 -0.207

1blg -o. 97096 +0.049

3a 18 -o. 97220 +0 .080 +0 .034

2a2u -0.98633 +0.197

2a18

-o. 99886 +0 .066 -0.040

1e -1.32625 +0.042 +0. 969 u +0. 969 -0.042

1a2u -1.39260 +0. 974

1a18

-2.37803 +0. 993 +0.019

•All values are in atomic units and were computed with basis set A, method M7 and at R=3. 72 a.u.

97

Table 6~5a One-electron energies of Ti 3+ and F-(separated ions)

Ti3+: Ct

-4.10178 -4.03134

-3.07947 -2.95762

-3.03964 -3.02223

-3.07947 -2.95762

-1.62694

2s

2px 2p

z 2p

y

a, a

-1.07469

-0.18122

-0.18122

-0.18122

Table 6.5b Same quantities~ of the SCF-salution for TiF~-

quantity 2T 2g

r. e:~ -14.15753 -14.05383 1 1

E. E:~ -13.80977 -13.71732 l. 1

tr(PH)** -463,94292 -463.96087

tr(QH)""' -445.12828 -445.16640

~tr (PGet) +224.89272 +224.95353

~tr (QGS) +215.65926 +215.72455

E scf

-468.51922 -468.44919

E rep +267.01043 +267.01043

E var -201.50879 -201.43876

orbital pop. of Ti:

(d +d +d )(l 1.03633 0.05969 xz yz xy (d +d +d >a 0.04608 0,05520 xz yz xy (d +d ) Ct 0.15966 1,08464 x2-y2 z2 (d 2 z+d zl 6 0,14886 0.15316

X -y Z

10Dq (present work) 0.07003 a.u. 1 • 906 eV 15,370 em 10Dq (ref. 1 ) 0.07975 2.170 eV 17,500 a.u. em exp

-1

-1

* Unless otherwise specified, the energy values are in atomic units, These data are computed with basis set A, method M7 and at R=3.72 a.u. Far the definition of the orbital population see eq. (6.3a-b).

**Gac and GSc (eq. 5.4) are included in the H-matrix (see also fig, 5.1).

98

Table 6.6 Evar and IODq* after each iteration-cycle

Evar(eq,2,16) Evu(eq.2.16a)

iteration-cycle 2'1' 2E 10Dq 2'1' 2E

2g 8 2s 8

0 -202.34467 -202.41425 -0.06958

1 -201.50673 -201.43579 +0,07094 -202,00685 -201.93977

-201.50876 -201.43872 +0.07004 -201.59537 -201.53714

-201.50879 -201,43875 +0.07004 -201.50687 -201.43920

-201.50879 -201.43875 +0.07004 -201.50905 -201.43887

5 -201.50879 -201.43875 +0.01004 -201.50888 -201.43882

6 -201.50879 -201.43876 +0.07003 -201,50883 -201.43878

•Atl valuee are in atomic units aud were computed with basis set A, method M7 and at R•3. 7 2 a. u.

10Dq

+0.06708

+0.05823

+0.06767

+0.07018

+0.07006

+0.07005

6.2 COMPUTED QUANTITIES WITH THE UNPROJECTED SINGLE DETERMI­

NANT WAVE-FUNCTION

6.2.1 Total electronic energy of the ground state 2

T2g and first

excited state 2

E g

The calculations indicated in table 6.1 are carried out 2 for the ground state T2 (d component) and first excited state

2 g xy 3-E (d 2 2 component) of the TiF6 complex. The computed values

g X -y of the total electronic energy Evar of the relating valence or-

3-bitals for a pure ionic mode~ of TiF6 are shown in table 6.7.

Table 6.7 E* for a var pure ionic model of TiF~-

2'1' 2E 2g g

R A B,C s* ,c* A s,c a* ,c*

2. 5000 -210.51042 -156.45645 -211.04902 -156.99505 3.0000 -204.94325 -150.87185 -205.19198 -151.12059

3.2500 -203.64390 -203.80702

3.4929 -202.84319 -148.76964 -148.76630 -20~.94922 -148.87567 -149.04343

3. 7200 -202.34467 -202.41425

3.8810 -148.01176 -148.06283

4.2691 -147.58473 -147.60793

-199.57956 -145.50273 -145.41174 -199.57856 -145.50273 -145.41174

. •All valu•a are in •tomie U1'1its.

These data are of course the same for and M7. From this table it can be seen

approximation method M4 that the energy of the

system becomes gradually lower with a decreasing metal-11gand

distance. Also, the total energy Evar is lower with basis set A (3s,3p included) than with basis sets B and c (3s,3p not includ­

ed), this is caused by the kinetic energy and nuclear-attraction energy of the 3s and 3p-orbitals.

99

The difference between the electronic energy computed with basis

set A and that based on the functions of set B and c, must be

constant. The deviation from this constant is caused by the ap­

proximations applied in the treatment of the core-repulsion term

which is different for both sets, because the size of the core

changes.

The computed values of the total energy after the first

iteration-cycle are assembled in table 6.8 for both methods of

approximation M4 and M7. Note that the energy values of the

Table 6.8 E:ar after the first iteration-cycle for various basis sets and at various metal-ligand distances

R A B c

2.5000 -204.77359 -154.19111 2T 3.0000 -149.84398 -150.75740 2g

M4 3.4929 -201.56905 -148.40993 -148.59758 -148.40583 -148.59238

(1) 3. 7200 -201.51706 3.8810 -147.86768 -147.90896 4.2691 -147.54679

2.5000 -205.15292 -154.88118 2E 3.0000 g -149.88157 -150.73448

M4 3.4929 -201.49185 -148.37982 -148.55850 -148.38574 -148.55497

(1) 3. 7200 -201.45766 3.8810 -147.84113 -147.88177 4.2691 -147.52889

2.5000 -201.81477 2T

2g 3.0000 -201.54140

M7 3.2500 -201.54504

(1) 3.4929 -201.53553 -148.58835 -148.59733 3.7200 -201.50673

3.8810 -147.91785 4.2691 -147.55494

2.5000 -201.15444 2E

g 3.0000 -201.29431

M7 3.2500 -201.38957

(1) 3.4929 -201.43304 -148.53606 -148.53366 3. 7200 -201.43579

3.8810 -147.88637 4.2691 -147.53569

•All values are in atomic units.

100

calculations with basis set C are lower than those with basis

set B. The results for the total electronic energy (basis set A) are

illustrated in fig. 6.2a~ for a detailed graph, see fig. 6.2b.

For the approximation method M4 the computed energy values are also given in the self-consistent solution~ see table 6.9. It

Table 6.9 It taken from the SCF-solution for various basis Evar

sets and at various metal-ligand distances

R A B c s• c•

2.5000 -207.96955 -156.00487 2T

2g 3.0000 -149.89694

M4 3.4929 -201.56949 -148,41240 -148,63522 -148.42229 -148.62120

(scf) 3. 7200 -201.51736 3,8810 -147.86848 -147.92176

4.2691 -147.55261

2.5000 -208.38588 -156.55013 2E

g 3,0000 -149,92339

M4 3.4929 -201.49227 -148.38103 -148,59052 -148.39378 -148.57831

(scf) 3. 7200 -201.45819

3.8810 -147,84179 -147.89250

4.2691 -147.53375

*All values are in atomic units.

has been found that, in contrast to the curve of the total ener­gy versus the metal-ligand distance for basis set A, the curves for basis sets B and C do not represent an energy-minimum nei­

ther for the ground state 2T2g nor for the first excited state 2

Eg for R > 2.5 a.u.

For basis set A, the theoretical equilibrium metal-ligand distance is evaluated by using a polynomial of the fourth de­gree. For the ground state this is 3.22 a.u. and for the first excited state 3.62 a.u. On the other hand, the experimental metal-ligand distance in TiF3 is 3.72 a.u.

The discrepancy between the results with basis sets A and

B is a consequence of the fact that the 3s and 3p-orbitals of the metal cannot be treated as core orbitals.

101

... "' >

LLJ

-200

-20 2Eg(excited state) ______ ...,.____

~-- __ .___

-203

-204

-205

-206

-207

-208 basis set A. method M 7 (1)

-209

r in a.u. ______.

Fig. 6.2a Total energy Evar as a function of the metal-ligand distance (for a detailed graph, see fig. 6.2b)

102

.45

c:

·5 5

3.0 3.1 32 rina.u.-·

Fig. 6.2b A detailed graph of fig. 6.2a

103

6.2.2 Crystal-field splitting parameter (IODq)

The crystal field splitting parameter or 10Dq follows

directly from the computed total energy of the excited state

E( 2Eg) and that of the ground state E( 2T28

}. It can be written as:

(6.1)

For the pure ionic modeZ indicated by (0), the values of the

10Dq-parameter are all negative (see table 6.10.). Table 6.10 also shows the effect of the approximated three- and four-centre two-electron integrals on the 10Dq-parameter. From this table it appears that the use of basis set A is to be pre­

ferred over that of basis set B or c. The first iteration-cycle is very important because then the

sign of 10Dq changes. After this cycle the influence of the it­eration process on the value of 10Dq is very small. Fig. 6.3 presents a few curves of the 10Dq-parameter as a func­tion of the metal-ligand distance.

3-The 10Dq-value for the TiF6 complex at experimental distance (3.72 a.u.) computed with basis set A turns out to be 0.07094 a.u. after the first iteration-cycle, and 0.07003 a.u. from the SCF-solution.

Evaluation of the 10Dq-parameter from the minimum energy of the ground state 2T2 {R=3.22 a.u.; E=-201.54513 a.u.) and that

g 2 of,the first excited state E

8 {R=3.62 a.u.; E=-201.43873 a.u.}

yields 0.10640 a.u. The experimental 10Dq is 0.0797 a.u. after Bedon et al.l (Fenske et al.3 proposed the value 0.0728 a.u.).

In literature a few variants of the conventional expression {eq. 6.1) for 10Dq are used. In these cases Koopmans' theorem4 is applied, assuming a constant orbital energy for all electrons with the exception of the promoted one (d ~ d 2 2). In this

xy x -y context equation (6.1) can be reduced to:

(6.2)

104

Table 6.10 Computed valuestof IODq for various basis sets and at various metal-ligand distances

M4 or M7

(0)

M4 ( 1)

M4 (scf)

M7

(1)

R A B c

2.5000 -0.53860 -0.53860 -0.53860

3.0000 -0.24873

3.2500 -0.16312

3.4929 -0.10603

3.7200 -0.06958

-0.24873

-0.16312

-0.10603

-0.06958

-0.24873

-0.16312

-0.10603

-0.06958

3.8810 -0.05107 -0.05107 -0.05107

4.2691 -0.02320 -0.02320 -0.02320

2.5000 -0.37933 -0.69007

3,0000

3.4929

3.7200

3.8810

4.2691

-0.03759

0.07720 0.03011

0.05940

0.02655

2.5000 -0.41633 -0.54526

3.0000

3.4929

3. 7200

3.8810

4. 2691

2.5000

3.0000

3.2500

3.4929

3. 7200

3.8810

4.2691

0.07722

0.05917

0.66033

0.24709

0.15547

0.10249

0.07094

-0.02645

0.03137

0.02669

0.02292

0.03908

0.02719

0.01790

0.04470

0.02926

0.01886

0.05229

0.03148

0.01925

tAll values are in atomic units

c•

-0.27713 -0.27713

0.02009 0.03741

0. 02851 0.04289

0.06367

For testing this equation we have assembled in table 6.11 the

105

one-electron energies after each R=3.72 a.u., basis set A and method tion the 10Dq-parameter is 0.09382 the 10Dq-value from equation (6.1),

iteration-cycle for the case:

M7. According to this equa­a.u., which is higher than

i.e. 0.07003 a.u.

106

0.7

as

Q4 ·.

U2

Q3 ·· .. ······•····••···· .....

t 0.1

··~~ .. ""-··~ ..

········ ............ . +

~ 0 c ,..,.-·--·-:.::::.::=-..:.==.::--=-·-·-·-....... ---·--+

..................

gO.O .5 . 3.0,.... ::? rina.u.- /'

-0.1

-0.2

-03

-04 I

-Q7

I I

I I

I I

I I

I

3S

- ionic model + basis set A,method M4(1l

--- .. .. B. .. M4(1l .. C. M4(1J .. A, M7m .. C. M7(1) 0

Fig. 6.3 IODq as a function of the metal-ligand distance

Table 6.11 Molecular-orbital energies of 3blg and 2b 1' 2g

2T 2E iteration 2g g 10Dq

cycle a a (l E(l(b

2g) (eq.6.2) e (b

28) E (bIg) E (bIg)

1 0.020571 0.747988 0.180055 0,663918 0.159484 2 0.020295 0.790020 0.115180 0.718428 0.094885 3 0.034937 0.804571 0.128547 0.739017 0.093610 4 0.034634 0.804050 0,128530 0.739188 0.093896 5 0.034685 0.804024 0.128523 0.739235 0,093838 6 0.034702 0.804014 0.128526 0.739256 0.093824

1' All values are in atomic units and were computed with basis set A, method M7 and at Rc3,72 a.u.

6.2.3 Orbital populations

The orbital population Pr of a basis function r is defined

by:

(6. 3a)

Pr(:! = ~ Q s L rs rs (6.3b) s

The index s is taken over all basis functions. We have computed the 3d-orbital populations with the basis sets A,B,c,B* and c*, they can be found in table 6.12. These values refer to the results after the first iteration-cycle and are the same for the approximation methods M4 and M7. The differences in the values figuring in the columns of table

6.12 (denoted by f pr) for the basis sets A and Bare caused by the fact that the off-diagonal elements of the F-matrix are not

equal for these two sets. This in its turn results from the treatment of the core which is different in both sets. The dif~ ferences in the values in the various basis sets A,B and C

columns denoted by t p for the s s are caused by the mixing of the

107

3dz2 orbital with the 3s (4s)-orbital.

Note that the values of ~ Pr and ~ ps increase as the metal­

ligand distance decreases.

With basis set A, the total population of the 3d-orbitals is at

the experimental metal-liqand distance (3.72 a.u.) 1.333 for the

qround state and 1.285 for the first excited state. The total

population of the 4s and 4p-orbitals is with basis set c for the

two electronic state 0.085 and 0.092 respectively.

Table 6.12' Orbital populations for various basis sets and at var­ious metal-ligand distancest

basis set

A

B

c

c*

R

2.5000

3.0000

3.2500

3.4929

3.7200

2.5000

3.0000

3.4929

1.355262

1.181164

1.132736

1.099338

1.076329

1.136368

1.087397

1.055565

0.844839

0.528706

0.408653

0.320746

0.257113

0.314255

0.279482

0.191380

3.8810 1.038975 0.138898

3.0000 1.087397 0.279494

3.4929 1.055565 0.191399

3.8810

4.2691

3.4929

3.4929

1.038975

1.027114

1.102109

1.102109

0.138909

0.102099

0.387033

0.387024

0.437007

0.222605

0.162878

0.121719

0.093407

0.171767

0.108657

0.068480

1.634458

1.396141

1.305801

1.239689

1.191892

1.237095

1.209645

1.143103

0.047808 1.103589

0.108657 1.209660

0.068480 1.143125

0.047808

0.033155

0.130603

0.130603

1.103602

1. 076031

1.291453

1.291441

; Distances are in atomic units. x Indicating the total population of the dxz•dyz dxy-orbitals.

Indicating the total population of the dxZ-y2• dz 2-orbitals.

108

6.2.4 Testing the Wolfsberg-Helmholz approximation

The non-empirical calculations on the TiF~- complex enable us to test the Wolfsberg-Helmholz approximationS often used in semi-empirical computations. The method supposes a constant k •.

lJ which may be employed for the evaluation of the off-diagonal elements F .. , its relation is represented by:

lJ

F •• + F •• F .. = k .. S.. 11

2 JJ

l.J l.J l.J (6. 4)

with k .. = 1.67 for a-bonding and k .. = 2.00 for ~-bonding. We lJ l.J

have evaluated the k .. -factors from the results F .. and F .. of a 3-l.J l.J 11

calculation on TiF6 with the following data: R=3.72 a.u.; basis set A, method M7, and the SCF-solution for the a-spin electrons in the ground state. The k .. -factors can be found in table 6.13.

l.J The same procedure has been repeated on the compos~ng parts of the F-matrix, namely the H-matrix and the Ga-matrix,and we found values fork!. and k~. which were very close to 1. The differen-

l.J l.J ces between the k .. -factors and the k!.-factors (or k~.-factors)

l.J l.J lJ are a consequence of the fact that the elements of the matrices H and Ga are great number with opposite sign. The computed k .. -values in table 6.13 do not eLucidate the as­

l.J sumptions of Wolfsberg and Helmholz.

6.3 COMPUTED QUANTITIES BEFORE AND AFTER SPIN PROJECTION

2 6.3.1 Average expectation value of the S -operator

The single determinant wave function of the unrestricted Hartree-Fock method is generally no eigenfunction of the s2

-

operator. As a consequence the average expectation value <82> is different from the number s'(s'+1) and we have to use the gener­al formula of <82

> shown in equation (3.10). The wave function

can be corrected by the single annihilator As'+l (see paragraph 3.1.2).

109

Table 6.13 Some k .. -values* from the Wolfsberg-Helmholz l.J

i

approximation

j

dx2_y2

dx2-y2 d xz dz2

dz2

dz2

dz2 d

3s

3s

xy

k •• l.J

7.1583 -4.0957

-1.8255 6.3254

-4.8152

6.1491

-5.1470 11 • 7749

1.8830

2.3717 2.3493

1. 7072 2. 0372

1.3605 2.0416 2. 1095 1 • 0411

1.8137

5.4386 4. 7361 4.2863

3.5753 -0.2175

k!. l.J

0.9750 1.0297 0.9400

0.9750 1.0297

0.9750 1.0297 0,9400

1.0496 1.0804 1.0737

1.0849 1.1229

1.0102 0. 9771 1.0425 1.1700 1. 0677

1.1350 1.1829 1.2363

0.9014 0.9356

k~'. l.J

0.9039 0.9648

0.8921

0.9046 0.9652

0.9045 0.9642

0.9128

0.9902 1. 0201 1 • 0165

1.0222 1 • 0331

0.9871

0.9390 1.0020 1.1785 1.0393

1.1023 1 .14 71 1.2056

0.8884 0.9411

*computed from the data of the SCF-solution of TiF~- at

R•3.72 a.u. (basis set A and method M7).

The expression for the expectation value <S2> after single

annihilation is given in equation (3.15). Computed values for

110

<82> before and after single annihilation are given in table

6.14 for the ground state and first excited state of the TiF~­complex with R=3.72 a.u., basis set A and approximation method

M7. As to the numbers in table 6.14 we observe that the <8

2 >-value

after single annihilation can in principle not be lower than

s'(s'+1), but these differences must occur as a consequence of

the assumption made in the derivation of <82> (see section as a

3.1.2).

Table 6.14 Computed values lll

of <82>

2T 2E 2g g

iteration- s' (s'+1) <52> <S2> <52> sd < S2> cycle sd a sa as a

1 0.750000 0.750308 0.750000 0.750312 0.750000 2 0.750000 0.750526 o. 750001 0.750464 0.750001 6 (scf) 0.750000 0.750650 0.750000 0.750576 0.749999

* with basis set A, method M7 and at R=3.72 Computed a.u.

We have computed also the <82 >-values for other basis sets (B,C)

as well as for other metal-ligand distances, but these numbers differ hardly from those in table 6.14.

6.3.2 Charge-density and spin-density functions

The expressions for the charge-density and spin-density functions before and after single annihilation are given in equations (3 .1Ba-b) and (3 .19.a-b) • By means of these equations we have computed the total charge density q(r) and total spin-

3- - . density p(£) in the XY-plane of the TiF6 complex as well as on the Ti-F axis. The graphs for these quantities after single

annihilation in the ground state 2T2g and in the first excited state 2

Eg are shown in the figures 6.4a-b and 6.5a-b. These diagrams illustrate the results obtained by a calculation

with R=3.72 a.u., basis set A and method M7. The computations show that, in contrast to the decreasing ab-

111

1.8

1.6 ,, I I I I I I

1.4 I 1-----'EJd,:z..~ component) \ I \

12 I \ I I I I ~T29 ld,1 component) 1.0 r J I

I I I 0.8 I J \

II \ 1/ \

0.6 IJ I v \

I 0.4 \

\ \

fo.2 \

:-.. '-.. q{x)

.... _ 00 2 3 4 5 t x ina.u.- t Ti F

Fig. 6.4a Charge-density functions on the metal-ligand axis

aoos 1 0.6 (\---

2Eg(d•'·•' component) t

I 0.005 r ~as ,

I I <dsy component> I I

~ 04 0.004 I I I I I

0.003 I ~ 03 I I I I

0002 I i02 I I J I I i 0.1 0001 I

I. I

f I

0 f 0

POO x inaJJ.- I P(IC)

-0001 {-01 I I

-0.0021 t J-0.2

Ti F

Fig. 6.4b Spin-density functions on the metal-ligand axis (dashed curve belongs to the dashed axis)

112

:i .. . 5

.,__

5

F yin au--.. i

2T2g (d1 ycomponent)

Fig. 6.Sa Charge-density functions in the XY-plane

5

solute-value of the spin-density function, the charge-density

function hardly change by single annihilation.

From the figures 6.4a and 6.5a we derived 1.85 a.u.(0.979 R) for

the radius of the F- ion and 1.87 a.u.(0,990 R) for the Ti 3+

ion. The radius of the F- ion may be compared to the value

1.16 R in the paper by Morris 7 ; Pauling gives 1.36 R.

The spin density ps(~) at the nuclei of Ti 3+ and F-are

calculated before and after single annihilation for the compo­

nents of both the ground state (d ) and the first excited state xy table 6.15. (d 2 2); see

X -y The spin density at the nucleus of the F-ion becomes after sing-

2 le annihilation -0,001417 for the ground state T2g and 0.117366

for the excited state 2E • The corresponding values of the fraa­g

113

TI F t yin au.- t ~~~~~=-~t-~-;~~~~~--~5

0»001

.... _

5

Fig. 6.5b Spin-density functions in the XY-plane

Table 6.15 Computed spin densities

Ti 3+ F-(nucleus 1) F - (nucleus 3) electronic

state

before after before after before after

2T 2g 0.018350 0.012235 -0.001226 -0.000817 -0.003926 -0.002618

2E 0.022286 0.014860 +0.177106 +0,176828 -0.002335 -0.001557 g

tional density f = P (O)/\x 2 (O) 12 with x2 (0) = -3.30863, be-

s s s • s come -0.000129 and 0.010721 respectively.

• No experimental data are available.

114

The assumption that the spin density at the F-nucleus is only due to the polarisation of the 2s electrons is based on the overlap (0,009378) of the 1s(F~) orbital with the metal (Ti3+)

3d-orbital which is much smaller than the analogous 2s(F-) over­lap (0.103052) 1 so that the polarisation of the 1s(F-) orbitals will be small, On the other hand, the value of x

15(0) = 14.5853

is considerably larger than the x25 (0) = -3.30863, Both consid­erations make this assumption doubtful.s

The isotPopia (contact) hypePfine paPameter can be computed with the help of the spin density p

8(0). However, the approach

in our calculations is not reliable enough for the determina­tion of a correct spin density at the F-nucleus (cf. ref. 9,10). To study the spin density in detail, we have to include the 1s orbital of the F- ion in the valence bases set, because this is the orbital with the highest density at the F-nucleus. It will be also recommended to take a smaller system, for example, the

.3+- 3+ 3-linear system T1 F Ti instead of TiF6 •

REFERENCES

1. Bedon, R.D., Horner, S.M., and Tyree Jr., S.Y., Inorg.Chem. 3, 647 (1964).

2. Siegel, s., Acta Cryst. ~. 684 (1956). 3. Fenske, R.F., Caulton, K.G., Radtke, D.D., and Sweeney, c.c.,

Inorg.Chem. ~. 951 (1966); ~. 960 (1966). 4. Koopmans, T.A., Physica I, 104 (1933). 5, Wolfsberg, M., and Helmholz, L,, J.Chem.Phys. 20, 837 (1952), 6. Freeman, A.J., and Watson, R.E., Phys.Rev.Letters ~. 343

(1961). 7. Morris, D.F.C., Structure and Bonding~. 63 (1968). 8, Pauling, L., The Nature of the Chemical Bond, Cornell

University Press, Ithaca, New York (1960). 9. Ellis, D.E., MIT Ph.D.Thesis (1966).

10. Ellis, D.E., and Freeman, A.J., J.Appl.Phys. ~. 424 (1968).

115

7 DISCUSSION

The unrestricted Hartree-Fock calculations on the octahe­

dral TiF!- complex, as described in this thesis, show that the computed values of the total energy as well as the 10Dq-parame­ter are extremely sensitive to both the treatment of the three­

and four-centre two-electron integrals and that of the core

electrons, the remaining molecular integrals are computed exact­ly. For example, the 10Dq-value with basis set A (for this nota­tion, see table 5.2) at a metal-ligand distance of 3.4929 a.u.

is 0.10249 a.u. if a~~ molecular integrals are included, and 0,04536 a.u. if the three- and four-centre two-electron inte­grals, except (A'B' llc'C') 1 are neglected. A part of this dif­ference, namely 0.02394 a.u., is caused by the influence of the

four-centre integrals (see table 5,3). The outcome of these calculations indicates that the four-centre integrals cannot be neglected,as was done by Shulman and Sugano1

in the non-empirical calculations on NiF:- and by Richardson2 on the transition-metal hexafluorides. Since we could not evaluate the three- and four-centre two-elec­tron integrals within a tractable computation time, we had to

approximate them with se~eated techniques. A method frequently used for this purpose is the Mulliken approximation (eq. 4.60). After applying it on the three-centre two-electron integrals, the 10Dq-value computed from the SCF-solution with basis set C and at a metal-ligand distance of 3.8810 a.u. is 0.07896 a.u. (the experimental 10Dq-value 3 is 0.07975 a.u.). On the other

hand, the total energy .(Evar) for the ground state 2T2g is

-166.60353 a.u., which is 21.10080 a.u. below the total energy of the separated ions (see table 6.7). This fact together with

the magnitude of the lowest one-electron energyt (-5.16 a.u.)

t The one-electron -1 .07469 a.u.

116

energy value of the 2s-electron in F is

indicates that the Mulliken approximation is not good enough for the three-centre two-electron integrals. From this example it will be obvious that a correct 10Dq-value in itself does not guarantee an equally good description of the other molecular quantities.

To obtain a better value for the total energy, the approxi­mation method indicated in equation (4.61) has been developed (on the test results of this equation, see table 4.5). The com­puted 10Dq-value and total energy are now 0.03283 a.u. and -147.62080 a.u. respectively. Following this approach, we have gradually included the remain­ing three- and four-centre integrals.

As to the treatment of the core electrons we found that a point-charge approximation for the core electrons in the two­centre interaction integrals (see the two-centre core error in the figs. 5.3a-d) raise the total energy by 0.30096 a.u. and the 10Dq-value by 0.00357 a.u. So it is obvious that a point-charge approximation in the two-centre Coulomb and exchange integrals yields such a large error in the 10Dq-value as well as in the total energy that this method is considered inadequate (see sec­

tion 5.3). This suggests that the results of Fenske4 are ques­tionable, because in the calculation of the diagonal elements of the matrix F, he used a point-charge approximation for aZZ elec­

trons.

The calculations with approximation method M4 (for this notation, see table 5.4) and at a metal-ligand distance of 3.72 a.u. yield in the SCF-solution with basis set C in compari-

2 son with basis set B for the electronic energy Evar( T28 > a value which is 0.1236 a.u.t lower and for 10Dq a value which is

t 0.0070 a.u. higher. The 10Dq-values resulting from the SCF-solution are with basis sets A and c and approximation method M7 0.07003 a.u. and 0.04011 a.u.t respectively. The great difference between these two numbers can be attributed to:

t Interpolated value.

117

(1) the point-charge approximation for the 3s and 3p core-elec­

trons in the off-diagonaL elements of H~

(2) the assumption that the 2s and 2p F-orbitals are orthogonal

on the core orbitals of Ti, being more closely approximated

by basis set A than by basis set C (see fig. 6.1a).

A calculation based on approximation method M7 and with a 3d

basis function having its charge-density maximum closer to the

ligands (basis set c*> gives in comparison with a calculation

using basis set C a higher value for 10Dq, a lower total energy

for the ground state and a higher total energy for the first ex­

cited state. Here,we do not refer to the results with method M4,

since these calculations contain more approximations than those

with method M7.

The calculations with basis sets A and B (or C) yield two

entirely different curves representing

the metal-ligand distance,

an energy-minimum at 3.22

namely the

a.u. for. the

component) and at 3,62 a.u. for the

the total energy versus

curve of basis set A has

ground state 2T (d 2g zXY

first excited state E8

(d 2 2 component) X -y while the curve of basis set B (or C) has no

energy-minimum at R > 2.5 a.u. A similar feature was found by 2 t 4-Richardson in a computation on FeF6 •

The total energy curve for the ionic model of TiF~- has,no ener-

gy minimum at R > 2.5 a.u. either.

The curve of the total energy {ground state) versus the

metal-ligand distance for basis set A and approximation method

M7 can be compared with that of the electrostatic model for

TiF~- (point-charge model) in which a Born repulsion (propor­

tional with r-n) is used in order to obtain the energy minimum.

In this context our results give a value for n = 5,5.

The value of 10Dq evaluated from the minima in the energy

curves is 0.10640 a.u. in comparison with 0,07003 a.u. deter­

mined as the difference between E( 2T2 ) and E( 2E ) at a metal-g g ligand distance of 3.72 a.u. (experimental distance in TiF3);

t Richardson used a Pestriated Hartree-Fock method and neglected all two-electron integrals with a two-centre charge distribu­tion consisting of two ligand (Slater) basis functions,

118

2 2 the quantity e(3b1 1 E8

) - e(2b28 , T28 ) is 0.09382 a.u. The 10Dq-values for TiF~- computed by Richardson are 0.0783 a.u.

with basis set (3d1 2s,2p) and 0.0676 a.u. with basis set

(3d,4s,4p,4dJ 2s,2p). The experimental 10Dq-value measured on

the compound NaK2TiF6 is 0.07975 a.u. and is presupposed to be 3-slightly different from the 10Dq-value of TiF6 • The influence

of the crystal field outside the cluster upon the 10Dq-value was

investigated by Ellis5' 6 on KNiF 3 and appeared to be about +3%.

It is interesting to note that the values of 10Dq(all nega­tive) for the pure ionic model at various metal-ligand distances are almost equal to the negative quantity of the corresponding 10Dq evaluated from the SCF-solution. Moreover, we found that both the .value of 10Dq and the total

number of electrons in the 3d-orbitals (evaluated according to the population analysis of Mulliken) increase with a decreasing

metal-ligand distance. We have studied this for R > 2.5 a.u. The charge on the Ti ion, as defined by Mulliken, is with basis

set A +2.61 in the ground state 2T2 and +2.65 in the first ex­cited state 2E • so the transition ~T2 + 2E brings about a ne-g g g gative charge transfer (0.04 a.u.) from metal to ligand.

From the values (tables 6.2 - 6.6) computed with basis set A, we see that all molecular orbitals of the electrons with a­

spin differ from those with a-spin, which is also valid for the relating orbital energies. The solution of the unrestricted

3-Hartree-Fock method is for the unpaired system TiF6 essentially different from that of the restricted method. It will be evident that the energy difference between the molecular orbital with a-

spin and that with a-spin decreases as the orbital is closer to the nucleus. For example, the energy orbital (presumably 3s) and the average

1a2u- and 1eu-orbital (presumably 3p}

difference of the 1a 1g­

energy difference of the turns out to be 0.07086

a.u. and 0.08589 a.u. respectively, which confirms the above statement.

Starting the SCF-iteration with a pure ionia model we found from the results in table 6.6 that the energy values computed with equation (2.16) instead of equation {2.16a) are always closer to the final SCF-energy. However, the change in the ener­gy value after each iteration-cycle indicates that the latter

119

equation is better than the former as regards the self consis­

tency of the problem. This is evident to a higher or lesser ex­

tent, because the evaluation of the total energy according to

equation (2.16a) -is directly related to the molecular orbital

energies and thus to the variations in it after each iteration­

cycle. Equation (2.16), however, requires the elements of the

density matrices P and Q which are summations of products Cikcjk

and contain therefore only an "average variation" of the coeffi­

cients Cik" The above-described self-consistency criterion has

not the disadvantages as those used in the semi-empirical cal­

culations, in which one defines (more or les arbitrarily) a

charge on the ions that must be constant (cf. ref. 7).

The upper-filled molecular orbital has an energy which is

positive by 0.03470 a.u. in the ground state and 0.12853 a.u.

in the excited state,

orbitals are unbound

which implies that the electrons in these

and that the free ion cannot exist. In an

environment of some positive ions, for example in the compound

NaK2TiF 6 , all one-electron energies will decrease with an almost

constant amount.

The difference between the computed equilibrium distance (3.22 · 3-a.u.) for the ground state of the isolated TiF6 cluster and the

experimental distance (3.72 a.u.) in TiF3 may be connected with

the covalency effects with the surrounding ions.

The computed values of

possibility of testing the

semi-empirical calculations

the matrices S and F give us the

approximation methods used in the

by Wolfsberg and Helmholz, 8 Ball-

hausen and Gray 9 and Cusachs. 10 ' 11 The values k .. , evaluated ac-l.J

cording to the former method, are represented in table 6.13. The

kij-values in this table do not maintain the assumption made by

Wolfsberg and Helmholz that k .. is a constant value, i.e. 1.67 l.J

for a-bonding and 2.00 for n-bonding. For the other two approxi-

mation methods we arrive at 'the same conclusion.

As mentioned before the molecular orbitals obtained by the

unrestricted Hartree-Fock method on TiF~- are different for the

a and s-set and therefore we expected a value of <82

> which dif­

fers more or less from s'(s'+1) in which s' is the total spin of

the system.

120

The value of <82> appeared to be 0.750650 for the ground state

2T2 and 0.750576 for the first excited state 2E ~ after single g g annihilation of the component with a spin (s'+1) the values of

<8 2> are 0.750000 and 0.749999 respectively. 2 The total wave function of the ground state T2 (d component) g xy

has a negative spin density between the Ti and F ref. 5 1 12) in contrast with the first excited state

component) which has no

6.4a and 6.5a).

negative spin-density region

nuclei (cf. 2E (d 2 2

g X -y (see figs.

The total fraationat spin densities f at each F-nucleus are for s

the two states -0.000194 and 0.010714 before and -0.000129 and

0.010721 after single annihilation. The computed spin densities p

5(£) and extremely dependent on the

accuracy of the matrices P and Q, and also on the basis set. For

example, the spin density at the Ti-nucleus is with basis set A positive (0.012235) contrary to the value (-0.009037) computed with basis set C (cf. ref. 13,14).

It is interesting to note that the computed spin density at the F-nucleus with basis sets A,B,c,a* and c* all have the same qualitative behaviour.

It was found that at a decreasing metal-ligand distance, the spin density at the F-nucleus becomes more negative for the

ground state and more positive for the excited state. Moreover, the spin-density values are different before and after

single annihilation in contrast with the charge density which is insensitive to this operation (cf. ref. 15). The ionic radii of the Ti3

+ and F- ion derived from the charge­

density function are 1.87 a.u._ (0.990 I) and 1.85 a.u. (0.979 I)­respectively.

The approximations used in our unrestricted Hartree-Fock 3-calculations on TiF6 are more or less hazardous for the compu-

tation of the spin properties such as the spin density and the hyperfine interaction, since the 1s-orbital of the F- ion is not included.

REFERENCES

1. Sugano, s., and Shulman, R.G., Phys.Rev. 130, 517 (1963). 2. Richardson, J.W., (private communication)-.--

121

3. Bedon, H.D., Horner, S.M., and Tyree Jr., S.Y., Inorg.Chem. 3, 647 (19.64).

4. Fenske, R.F., Caul ton, K.G., Radtke, D. D., and Sweeney,c·.c., Inorg.Chem. 5, 951 .(1966).

5. Ellis, D.E., MIT ih.D. Thesis (1966). 6. Ros, P., Ellis, D.E., and Freeman, A.J., Optical Properties

of Ions in Crystals, p.231 (H.M. Crosswhite and H.W. Moos, eds.),Interscience Publishers, New York, London and Sydney (1967).

7. Cusachs, L.C., and Politzer, P., Chem.Phys.Letters l• 529 (1968).

8. Wolfsberg, M., and Helmholz, L., J.Chem.Phys, 20, 837 (1952)• 9. Ballhausen, C.J., and Gray, H.B., Inorg.Chem. T;" Ill (1962).

10. Cusachs, L.C., J.Chem.Phys. 43, 1575 (1965),-II. Cusachs, L.C., and Cusachs, B:B., J.Phys.Chem. 71, 1060

(1967). -12. Ellis, D.E., and Freeman, A.J., J.Appl.Phys. 39, 424 (1968). 13. Watson, R.E., and.Freeman, A.J., Phys.Rev.Letters 6, 277

(1961). -14. Freeman, A.J., and Watson, R.E., Phys.Rev.Letters !• 343

(1961). 15. Amos, A.T., and Snyder, L.C., J.Chem.Phys. ~. 1773 (1964).

122

8 CONCLUSIONS

The unrestricted Hartree-Fock calculations on the pure oc-3-tahedral TiF6 complex are carried out with a symmetry-unre-

stricted basis set. The resulting eigenfunctions (molecular or­bitals or one-electron wave functions) are classified according

to the irreducible representations of the point group n4h.

As to the expansion method (see paragraph 4.1.2) for an or­

bital around another centre we conclude that this does not give a correct description of the radial part R(r) around the nucleus

of the F ion if less than 30 terms are taken into account. How­ever, the values of the integrals evaluated by a method which

uses this expansion have 5 to 6 significant figures if 15 terms are involved.

The calculations with various integral approximations for the three- and four-centre two-electron integrals (all other molecular integrals are computed exactly), with various basis

sets for the metal ion and at various metal-ligand distances, yield a series of numerical quantities for the total energy, the crystal-field splitting parameter (1.0Dq) and the charge-density

and spin-density functions. With respect to these numbers we conclude:

(1) the computed total energy and 10Dq-value are very sensitive to the approximation method for the three- and four-centre

two-electron integrals. Using the Mulliken approximation for the three-centre two-electron integrals we arrive at a total energy which has no real physical meaning. On the other hand, the approximation method of equation (4.61) gives more reliable results for both the total energy and

10Dq7 (2) the total energy and 10Dq-value are dependent on the treat­

ment of the core electrons. So a point-charge approximation

for the core electrons is definitely u~acceptable7

123

(3) the calculations with basis set (3d) or (3d,4s,4p) for Ti3+

have at R > 2.5 a.u. no minimum in

the metal-ligand distance for both

and the first excited state 2E in g

the total energy versus 2 the ground state T2g

contrast with the basis set (3s,3p,3d). This feature is a consequence of the fact that in the first two calculations the 3s and 3p-orbitals are included in the core. The computed metal-ligand equili­

brium distance in TiF~- is in reasonable agreement with the experimental distance in TiF3 ;

(4) the value of 10Dq computed with basis set (3d,4s,4p;2s,2p) is greater than that with basis set (3d;2s,2p), while the result of basis set (3s,3p,3d;2s,2p) is still greater. The

latter fact is a consequence of the approximations for the core electrons. If in the basis set (3d,4s,4p;2s,2p} the 3d wave function is replaced by another 3d-function which has its electron density closer to the ligand, the value of 10Dq is larger. This fact can be attributed to an increase of the covalency in the calculation with the latter orbital in comparison with the covalency in that of the former;

(5) for R > 2.5 a.u. the magnitude of 10Dq as wel as of the

total population of the 3d-orbitals increases if the metal­ligand distance decreases.

(6)

All 10Dq-values of the pure ionic model at R > 2.5 a.u. are negative and their absolute value is almost equal to the 10Dq-value derived from the SCF-solution; with the transition 2T2 +

2E a small quantity of negative g g charge moves from the metal to the ligands;

(7) after each iteration-cycle the total energy evaluated with

equation (2.16) is closer to the final (SCF) total energy than the quantity resulting from equation (2.16a}. However, the changes occurring in the latter total energy are a better indication for the self-consistency of the total

wave function than the values yielded by equation (2.16).

Starting the iteration procedure with an ionic Hamiltonian we obtain after the first iteration-cycle a value of 10Dq which hardly changes in the subsequent cycles;

(8} the upper-filled molecular orbital has a positive energy,

which implies that the free TiF~- complex cannot exist;

124

(9) using the computed elements of the matrices S and F we cannot maintain the constant k .. in the approximation meth-

1J od of Wolfsberg-Helmholz, Ballhausen-Gray and Cusachs. On the other hand, the same procedure applied on the elements of the matrix Hand Ga(GS) yield a k!.-value which is very·

1J close to 1,

2 (10) the average expectation value of the S -operator is very

close to the number of a pure spin state with a spins'. Single annihilation of the spin component with a spin

(s'+1) changes the value in that corresponding to a pure spin state,

2 3-(11) the ground state T28 of TiF6 has a negative spin density

between the Ti ion and the F ions. The fractional spin den­sity f

8 at the F nucleus is negative. On the other hand,

the first excited state 2E has a positive spin density all g

over the complex (f8

on the F nucleus is also positive);

(12) the spin-density at the nucleus is very sensitive on the

basis set used for the ion in question, but hardly depen­dent on the basis set applied to the neighbouring ions. In contrast with the charge-density function, the spin­density function changes considerably by single annihila­tion. When the metal-ligand distance decreases, the £

8-

values on the F ion change slightly in the sense of ~ecom­

ing more negative in the ground state and more positive in the first excited state.

3-This study on TiF6 shows that several physical properties

can be evaluated without using any empirical parameter, but it will also be obvious that the approximations used in the calcu­lation must be chosen very carefully.

125

c~ (c~) -L -L

c!l.(lAmA;lA_mA_)

f s

h (1) , h (12)

i (X) p

k (x) p

n,l,m and s

p,q

q(f)

1"12

s'

u(r ,R)

A,B,C A,B,C,D

LIST OF PRINCIPAL SYMBOLS

ith eigenvector of an electron with a(B)-spin

Gaunt coefficient

fractional spin density

one- and two-electron Hamiltonian operator

spherical Besselfunction of the first kind

spherical Besselfunction of the second kind

quantum numbers

number of electrons with a and B-spin respectively

orbital population of basis function r with a(B) spin

charge-density function

distance between electrons 1 and 2

total spin It J\+1

r<lr> with r< the lesser of r and R and r> the greater of r and R

p-orbitals at nucleus P

basis sets nuclear indices; orbital on centres A,B,C and D

single annihilator

AP(x), s,<x)} auxiliary functions

1!: (X) p

ca (CB) matrix of eigenvectors (columns)

total energy

Hartree-Fock matrix

127

H

J

J' j (1)

Ka (K$)

Kp+l(x)

K. (1) J

N

p (Q)

Plm (x)

plm(x)

RAB

Rnl (r)

Rnl (r)

s

slm(a,;j>)

UA(2)

vnlmp(t,r)

128

Hartree-Fock operator

total electronic interaction matrix

total electronic interaction operator

one-electron matrix

total Hamiltonian operator

Besselfunction of the first kind with imaginary argument and half integral order

"Coulomb" matrix

"Coulomb" operator

"exchange" matrix

Besselfunction of the second kind with imaginary argument and half integral order

"exchange" operator

total number of electrons

spin projection operator

density matrices for electrons with a(B)-spin

unnormalised associative Legendre function

normalised associative Legendre function

distance between centres A and B

unnormalised radial part of the wave function

normalised radial part of the wave function

overlap matrix

spin operators

linear combination of spherical harmonics

potential function

auxiliary function for expanding an orbital around another centre

spherical harmonics

nuclear charge of atom A

n,a

a,a,y

I;;. l.

p(f:)

'i (1)

X ~A) (1) l.

1/J i ( 1 )

-~ll

«>m (')

'!'

!"!~~B) l.J

spin functions

Eulerian angles

energy of ith molecular orbital

ith exponent in the radial part of an orbital th spin function of i molecular orbital

spheroidal coordinates

spin-density function th i molecular orbital, occupied by electron 1

ith atomic orbital, occupied by electron 1 and centred at nucleus A

ith molecular spin-orbital,occupied by electron 1

kinetic-energy operator

normalised ~-part of the wave function

total wave function

charge distribution around centres A and B

129

SUMMARY

Dur'ing the last few years several research groups have

tried to find an adequate quantum mechanical description on

transition-metal complexes in order to get an insight into the electronic structure and the relevant physical quantities.

None of the non-empirical calculations edited sofar, however, succeeded in ascertaining an applicable numerical value of the

crystal-field splitting parameter (10Dq) of the complex in ques­

tion.

This thesis deals with a symmetry- and spin-unrestricted

Hartree-Fock method, using a single determinant wave function. To this end, the conventional Hartree-Fock method is used as a starting point; the unrestricted Hartree-Fock method being ar­rived at after elimination of the spin- and symmetry-restric­

tions. A spin projection method (single annihilation) is dis­cussed as well.

Formulae and computer programmes are developed for the

molecular integrals required; the basis functions in question are of the Slater-type.

The symmetry- and spin-unrestricted Hartree-Fock method is 3- I applied to the octahedral TiF6 complex (d -system).

The influence of both the three- and four-centre two-electron

integrals and the treatment of the core-electrons on the quanti­ties to be computed is discussed. Various approximation methods for these integrals are developed, because the exact computation would still require too much computer time. Next, calculations are carried out with various basis sets and at various metal-ligand distances, in which computed values are given for the total energy of the ground state 2T2g as well as that of the excited state 2Eg, the crystal-field splitting pa­rameter (10Dq), the orbital population, the average expectation

131

value of the s2-operator and the charge-density and spin-density functions. Furthermore, the effect of a spin projection on the total wave function with respect to the latter three quantities is studied. The computed elements of the Hartree-Fock matrix together with the overlap matrix make it possible to verify the (more or less

arbitrarily) assumed parameters used in the semi-empirical meth­ods.

The calculation method developed in this thesis leads to a -1 . 3-10Dq value (15,370 em ) for TiF6 which matches quite well the

-1 experimental value (17,500 em ) in the compound NaK2TiF6 • The computed metal-ligand equilibrium distance in the ground state 2T (1.70 ft) and that in the first excited state 2E (1.92 ft) 2g g come close to the experimental distance (1.97 ft} in the solid

TiF3• The radii of the Ti3+ and F- ions in the TiF!- complex de­

rived from the charge-density function are 0.990 K and 0.979 R respectively.

132

SAMENVATIING

Gedurende de laatste jaren hebben diverse onderzoekers ge­

probeerd voor de overgangsmetaal complexen een quantum-mechani­

sche beschrijving te vinden om daardoor een inzicht te ver­

krijgen in de elektronenstruktuur en de bijbehorende physische

grootheden. De niet-empirische berekeningen, die gepubliceerd

zijn, waren echter geen van alle in staat de numerieke waarde

van de kristalveld-parameter 10Dq van het desbetreffende complex

goed te beschrijven.

In dit proefschrift wordt een symmetrie- en spin-unrestric­

ted Hartree-Fock methode besproken, waarbij gebruik gemaakt

wordt van een determinant-golffunktie. Hiertoe wordt uitgegaan

van de conventionele Hartree-Fock methode, waarbij na weglaten

van de symmetrie- en spin-restricties de unrestricted Hartree­

Fock methode wordt verkregen. Eveneens wordt een spinprojectie

methode (enkelvoudige annihilatie) toegelicht.

Formules en computerprogramma's worden ontwikkeld voor de

vereiste moleculaire integralen1 de basisfunkties hierin zijn

van het Slater-type.

De symmetrie- en spin-unrestricted 3-wordt toegepast op het octaedrische TiF 6

De invloed op de te berekenen grootheden

Hartree-Fock methode

complex (d 1-systeem).

zowel van de drie- en

vier-centra twee-elektron integralen als van de behandeling van

de core-elektronen wordt nagegaan. Verscheidene benaderingsme­

thoden voor deze integralen worden ontwikkeld, daar de exacte

berekening thans nog te kostbaar is.

Vervolgens worden berekeningen uitgevoerd met verschillende

basis-sets en verschillende metaal-ligande afstanden, waarbij de

navolgende grootheden worden bepaald: de totale energie van de

grondtoestand 2T en die van de aangeslagen toestand 2E , de 2g g

133

kristalveld-parameter 10Dq, de orbital populatie, de gemiddelde

verwachtingswaarde van de operator s2 en de funkties van de

ladingsdichtheid en sp1nd1chthe1d. Bovend1en wordt het effect

nagegaan van een spinprojectie op de totale golffunktie met be­

trekking tot de drie laatstgenoemde grootheden.

De berekende elementen van de Hartree-Fock matrix en die van de

overlap-matrix geven de mogelijkheid de (min of meer willekeu­

rig) aangenomen parameters, die gebru1kt worden in de semi­

empirische methoden, op hun betrouwbaarheid te onderzoeken.

De in dit proefschrift ontwikkelde rekenmethode geeft een -1 3~

10Dq-waarde (15.370 em ) voor TiF6 , die goed overeenkomt met

de experimentele waarde (17.500 cm-1 > in de verbinding NaK2TiF6 •

De berekende metaal-ligande evenwichtsafstand in de grondtoe­

stand 2T2g (1,70 ft) en in de eerste aangeslagen toestand 2Eg

(1,92 ft) benaderen de experimentele afstand (1,97 i) in de vaste 3+ - 3-stof TiF3 .De ionstralen van Ti en F in het TiF6 complex zijn

respectievelijk 0,990 ft en 0,979 R.

134

LEVENSBESCHRUVING

Franciscus, Lambertus, Martinus, Arnoldus, Henricus de Laat

werd op 19 juli 1942 geboren te Nuenen. Hij bezocht de St. ~loysius MULO te Eindhoven en behaalde in 1957 de diploma's MULO A en B. Vervolgens verwierf hij in 19641

te Eindhoven het HTS-diploma (afdeling Chemie). Tijdens deze op­leidinq verrichtte hij in 1959 een stage bij het Studiecentrum voor de Kernenergie (CEAN) te Mol-Donk (Belgil!).

In oktober 1961 werd hij als student ingeschreven aan de Technische Hogeschool te Eindhoven (afdelinq der Scheikundige Technologie), waar op 12 juli 1962 het propaedeutisch examen, op 18 december 1963 het kandidaats examen en op 18 november 1964 het doctoraal examen werden afgelegd. In 1965 verkreeg hij de onderwijsbevoegdheid in Natuur- en Scheikunde. Van 1965 tot 1967

was hij als part-time docent werkzaam bij het middelbaar onder­wijs. Thans is hij wetenschappelijk medewerker bij de Stichting Schei­kundig Onderzoek Nederland (ZWO).

Het onderzoek, in dit proefschrift beschreven, werd gestart in november 1964. Het grootste gedeelte van de berekeningen ward mogelijk gemaakt door de installatie van de EL-XS computer op de Technische Hogeschool te Eindhoven in november 1966,

De deelname in 1964 aan het "NATO Advanced Study Institute in Theoretical Chemistry" te Frascati (!tali~) en in 1967 aan de "Summer School in Theoretical Chemistry" te Oxford (Engeland) leverde een waardevolle bijdrage tot zijn theoretische kennis.

135

STELLINGEN

1. In de publikaties betreffende photodesorptie van zuurstof aan NiO wordt ten onrechte geen aandacht besteed aan de desorptie van zuurstof als gevolg van de temperatuurverho­ging van het NiO poeder tijdens het belichten.

Uaber, J., en Stone, F.S., Trans.Faraday Soc. 59, 192 (1963). Thomas, J.M., en Thomas, W.J., Introduction to the Prineipl·es of Heterogeneous Catalysis, Academic Press, London, New York (1967), p. 287. DeLaat, F.L.M.A.H., Internal Report of Inorg. Chem. Dept., Technological University, Eindhoven (1964).

2. De door Fenske en Radtke gevolgde methode ter bepaling van de kristalveld-parameter 10Dq is onjuist.

Fenske, R.F., en Radtke, D., Inorg.Chem. z, 479 (1968). Dit proefschrift.

3. De konklusie, getrokken uit de radiochemisch bepaalde akti­viteiten van 192Ir en 36c1 in papierelektroforetisch ge-

.. 3-scheidea:bydrolyseprodukten van 0,002 molair gemerkt IrC16 in 0,001 normaal zwavelzuur, is theoretisch aanvechtbaar.

Van Ooy, W.J., en Houtman, J.P.W., Radiochimica Acta z, 118 (1967).

4.. Het valt: te betwijfelen of alle door Van Belle bepaalde waarden van dipoolmomenten in verdunde oplossingen juist zijn.

Van Belle, o.c., Proefschrift, Leiden (1967).

5. De resultaten door Hoffmann verkregen bij zijn berekenin­gen aan positieve en negatieve ionen zijn aan twijfel onder­hevig, daar geen rekening wordt gehouden met de afhanke­lijkheid tussen de diagonaal matrix-elementen en de lading van het betreffende atoom.

Hoffmann,. R., J.Chem.Phys. 39, 1397 (1963). Hoffmann, R., J.Chem.Phys. 40, 2480 (1964).

6. De stationaire fase Ucon SOHB wordt door Leibnitz en Struppe beschreven als een polyethyleenglycol met een gemiddeld molekuulgewicht van 2000. Dit is niet verenigbaar ~t~ gaschromatografische eigenschappen van deze fase.

Leibnitz, E., en Struppe, H.G., Handbuch der Gas-Chro­matographie, Geest en Portig K.-G., Leipzig (1966), p. 365. McReynolds, W.O., Gas Chroma~ographic Retention Data, Preston Technical Abtracts Company, Evanston, Illi­nois (1966), p. 172.

7. De konklusie van Offenhartz, dat de berekende kristalveld­par~ter 10Dq met gebruikmaking van de atomic orbitals van de gescheiden ionen als basis funkties niet zal overeenstem­~n met de experimentele waarde, is onjuist,

Offenhartz, P.O., J.Chem.Phys. Dit proefschrift.

• 2951 (1967).

8. De twee ESR signalen door KrUerke en Jung in X en Y zeolie­ten aangetoond, zijn niet in overeenstemming met de aann~ dat er analoog aan de ionen ca2•, sr2• en Ni 2+ voor cu2•

eveneens vier verschillende sites ter beschikking zouden zijn.

KrUerke, U., en Jung, P., Z.physik.Chem. Neue Folge 58, 1 (1968). Olson, D.H., J.Phys.Chem. 72, 1400 (1968).

9. De algemene veronderstelling, dat het menselijk prestatie­niveau op atletiekgebied is begrensd, komt niet overeen met de aard van de prestatieverbeteringen gedurende de laatste vijftig jaar.

Eindhoven, 5 november 1968 F.L.M.A.H. de Laat