Lecture Notes in Chemistry Edited by G. Berthier, M. J. S. Dewar, H. Fischer, K. Fukui, H. Hartmann, H. H. Jaffe, J. Jortner, W. Kutzelnigg, K. Ruedenberg, E. Scrocco, W. Zeil
1
Georges l-lenry Wagniere
Introduction to
Elementary Molecular Orbital Theory and to Semiempirical Methods
Springer-Verlag Berlin' Heidelberg· New York 1976
Author Georges Henry Wagniere Physikalisch-Chemisches Institut der Universitat Zurich RamistraBe 76 CH-8001 Zurich
Library of Congress Calaloging in Publica lion Dala
Wagni~re, Georges Henry, ~933-Introduction to elementary molecular orbi taJ. theory
and to semiempiricaJ. methods.
(Lecture notes in chemistry ; v. 1) Bibliogra.phy: p. Includes index. 1. Molecular orbitaJ.s. I. Title.
Q.D46~. W33 541' .28 76-4000e
ISBN-13: 978-3-540-07865-4 001: 10.1007/978-3-642-93050-8
e-ISBN-13: 978-3-642-93050-8
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© by Springer-Verlag Berlin . Heidelberg 1976
Softcover reprint of the hardcover 15t edition 1976
Introduction
These notes summarize in part lectures held regularly at the University of Zurich and, in the Summer of 1974, at the Seminario Latinoamericano de QUimica Cuantica in Mexico. I am grateful to those who have encouraged me to publish these lectures or have contributed to them by their suggestions. In particular, I wish to thank Professor J. Keller of the Universidad Nacional Autonoma in Mexico, Professor H. Labhart and Professor H. Fischer of the University of Zurich, as well as my former students Dr. J. Kuhn, Dr. W. Hug and Dr. R. Geiger.
The aim of these notes is to provtde a summary and concise introduction to elementary molecular orbital theory, with an emphasis on semiempirical methods. Within the last decade the development and refinement of ab initio computations has tended to overshadow the usefulness of semiempirical methods. However, both approaches have their justification. Ab initio methods are designed for accurate predictions, at the expense of greater computational labor. The aim of semiempirical methods mainly lies in a semiquantitative classification of electronic properties and in the search for regularities within given classes of larger molecules.
The reader is supposed to have had some previous basic instruction in quantum mechanics, such as is now offered in many universities to chemists in their third or fourth year of study. The bibliography should encourage the reader to consult other texts, in particular also selected publications in scientific journals.
I wish to express my gratitude to Miss H. B6ckli who has competently typed the entire manuscript and to Mr. E. Spalinger for the drawings.
Zurich, May 1976
G. Wagniere
Contents page
I. The hierarchy of approximations 1
1. The Born-Oppenheimer approximation 2. The solution of the electronic problem 3. The subdivision of electrons into different groups
II. Simple Htickel theory of ~ electrons 4
1. The LCAO-MO formalism 2. Further simplifications 3. Some important definitions
III. Many-electron theory of ~ electrons 15 1. Ethylene as two-electron problem 2. The configuration interaction (CI) procedure 3. ~e semiempirical PPP approximation for ~ electrons 4. Benzene as an example 5. Electric-dipole transition probability
IV. Self-consistent-field (SCF) methods 42 1. Simple LCAO-formulation of the closed-shell case 2. Semiempirical simplification (ZDO approximation) 3. More general formulation of the closed-shell case 4. Koopmans' "theorem" 5. Some remarks on localized orbitals 6. Open-shell SCF methods
6.1. The restricted open-shell SCF method 6.2. The unrestricted open-shell SCF method
V. All-valence MO procedures 59 1. The Extended Htickel (EH) method 2. Electronic population analysis 3. Semiempirical all-valence calculations,
including electron interaction
3.1. The CNDO (complete neglect of different~al overlap) method
4. Invariance of the SCF eigenvalue problem to unitary basis transformations
VI. Special topics 71 1. Optical activity 2. Selection rules for electrocyclic reactions
and cycloaddition reactions 3. Molecular orbital theory with periodic (cyclic)
boundary conditions
References
Subject index 100
I. The hierarchy of approximations
The nonrelativistic Hamiltonian for a molecular system com
posed of many nuclei (indices A,B) and many electrons (in
dices ~,v) reads (neglecting magnetic interactions):
I (- h2 2) --v 2me ~ +I(-~~AvX)
~ A
I I ZAe 2 + I I
e 2 I I ZAZBe 2
-- + r~A r~'V RAB
~ A ~>'V A >B
which may be more concisely written
Here T stands for the kinetic energy operator, V for the potential energy operator, the subscript e means "elec
tronic", the subscript n means "nuclear".
1. The Born-Oppenheimer approximation
We seek to solve the time-independent molecular Schrodinger equation
:lC '\f(r,R) '" W • f(r,R)
Due to the great mass difference between electrons and ato
mic nuclei it proves possible to a satisfactory degree of
approximation (1] to treat the degrees of freedom of the
electrons, designated collectively by r, separately from
those of the nuclei, designated here by R. In this sense
the solution f(r,R) may approximately be written as a pro
duct of two functions, of which one depends only on the
general nuclear coordinates R:
2
1[1 (r,R) ~'(r,R) • vCR)
By neglecting some terms which in general may be shown to
be small [1,2J, it is thus possible to separate the
Schrodinger equation into:
a) An equation for the motion of the electrons, the nuclei
remaining fixed at frozen positions R':
:J{ 1Ir (r,R') e m
Here m denotes a particular electronic state. The electronic
energy Em(R') depends parametrically on the frozen positions
of the nuclei. Often one holds the nuclei fixed in experi
mentally known equilibrium positions.
b) An equation for the motion of the nuclei in the field of
the electrons in a given electronic state m:
e . mJ
The electronic energy as a function of nuclear position Em(R)
acts as a potential on the nuclei. For a diatomic molecule
in a bonding electronic state Em(R) + Vnn(R) is generally
described by a Morse potential. vmj(R) represents a vibra
tional wavefunction j in the electronic state m.
2. The solution of the electronic problem
We consider the electronic equation
= (T + V + V ) 1jJ (r) e en ee m
We no longer explicitly mention the nuclear coordinates R,
once we have stated where they have been fixed. Of course we
still have a many-body problem to deal with, and the solution
of this problem is in general still very cumbersome. In
practice it proves only feasible to obtain approximate
3
solutions, and it is the degree of approximation that is the
crucial question. Even in simplest cases exact solutions re
quire a quasi-infinite amount of labor. In this sense we
distinguish between:
a) The ab initio procedure. It seeks in principle exact
solutions. All quantities appearing in the calculation are
computed as exactly as numerically possible. If an ab
initio solution is still approximate, which in practice it
always is, this comes from the fact that the form of the
wavefunction has been restricted to facilitate the compu
tation.
b) The semiempirical procedure. It seeks from the start
only approximate solutions. The simplifications may be quite
drastic, but must always be physically justifiable. One may
in this sense further distinguish between
i) simplification of the electronic Hamiltonian ~ itself
by, for instance, leaving out the electronic repulsion
term Vee and replacing Te+Ven by an effective Hamilto
nian;
ii) neglect of some intermediate quantities or their em-
pirical calibration on atomic data and on test-molecules.
To study large molecules procedure b) is often the only
tractable one. The more limited reliability of b) as com
pared to a) is sometimes also compensated by an increased
insight into the interrelation of basic quantities.
3. The subdivision of electrons into different groups
From the chemical point of view the electrons in a molecule
may be subdivided into those which take part in the forma
tion of chemical bonds, and those which are largely un
affected by bond formation. The former are generally called
valence electrons, the latter atomic core electrons. If the
4
molecule in its equilibrium conformation (i.e. the equilibrium
geometry of the atomic nuclei) possesses certain elements of
symmetry, such as for instance a plane of symmetry in which
lie all atoms of the molecule, the valence electrons may
appropriately be further subdivided into cr and ~ electrons.
From his experience the chemist knows that. this subdivision
is also physically meaningful. The presence of such ~ electrons
in a molecule influences decisively its reactivity and its
spectroscopic properties.
II. Simple HUckel theory of ~ electrons [3J
The electronic Hamiltonian
may be written
From it we split off a ~ electron Hamiltonian
The cr electrons and the nuclei are assumed frozen into a
molecular core:
(~) (~)
I hcore (tJ.) + I rtJ.\} tJ. >\1
We further average v~~ to obtain an effective ~ Hamiltonian
as sum of pseudo-one-electron parts:
0Uckel ~
(7r)
I heff(tJ.)
5
In this approximation each ~ electron moves in an average
~ of the core and the other ~ electrons.
We now want to solve the one-electron equation
€ • qJ
As we no longer have an explicit Schrodinger equation, the
solutions depend strongly on the form which we impose on them.
1. The LCAO-MO formalism
(and the Ritz variational principle)
We expand our one-electron functions or molecular orbitals
(MO) as linear combinations of basis functions or atomic
orbitals (AO) Xp and write
N
cP I CpXp p=l
In our present case the Xp are 2p~ (E 2pz) atomic orbitals
centered on each atom contributing one (or possibly two)
electron(s) to the ~ system. The total number of such atoms
~ N and the index p also denotes a given atom. The expectation
value for the one-electron energy € then takes the form
SCP*hqxlT €
Jcp*qxi T LL p q
The integrals over the AO's are abbreviated as indicated.
We then make use of the variational principle (without proof):
By making the energy a minimum with respect to the coefficients
c; or equivalently cq , the energy tends towards the lowest
eigenvalue: €min ---. eo. Necessary conditions for a minimum
are:
6
o
(We assume these conditions for our purposes also to be
sufficient. )
We write:
and differentiate with respect to
~:* (I I C;CqSpq) + E: (I Cq8pq ) P P q q
where p = 1,2 ••.• N. Setting the
* c • p'
I q
Cqhpq
derivatives ~ equal to bc*
zero, we obtain the following equations: p
C1 (h11 - eS11 ) + c2(h12 - eS12) +
C1 (h21 - eS21) + c2(h22 e822) +
+ cN(h1 N - eS1 N)
+ cN(h2N- eS2N)
This is a system of N linear homogeneous equations with
o
o
N + 1 unknowns, namely N coefficients c1 ..• cN' and the
eigenvalue e. These equations have non-trivial solutions only
if the determinant of the coefficients vanishes:
o
This determinant is also called the secular determinant, its
polynomial expansion the secular equation.
7
There will, in general, be N solutions for e • To each eigen
value en there corresponds an eigenfunction ~n' To obtain the
coefficients cnp the condition of normalization must also be
invoked:
1
The solutions then are:
eigenvalues eigenvectors
q:>1 = r C1 pXp P
~2 r C2PXp p
coefficients
Strictly speaking, from a variational point of view only
the lowest solution is physically admissible. If the molecule
of interest has certain elements of symmetry and the solutions
transform according to different irreducible representations,
then the lowest solution of each irreducible representation
is admissible. In general, however, in the frame of the
adopted crude approximations of the RUckel method, all solutions
are considered meaningful.
2. Further simplifications
We write hpp = a p and call it a coulomb integral
hpq = ~pq and call it a resonance integral.
We neglect resonance integrals, except between nearest
neighbors. We neglect overlap integrals; this corresponds
to the zero differential overlap approximation.
Example: Ethylene ----0 0-~&-8~
2
8
The secular equation is obtained as
I a-£ 13 I 13 a-£
and leads to the
I
" I I
I
----~ a a \ \ , , ,
U
o . • 2 (a-e) = 13
solutions { £1
£2
a-j3 e2 ((l2
a+j3 £1 1:P1
a + 13 a - 13
1 = 'f2' (X1 -X 2 )
1 V2' (X1+x2)
Figure 1 HUckel energy levels in ethylene
;;; 71*
;;; 71
Physically, a may be assumed to correspond to the energy of
an electron in a 2P71 orbital of an sp2-hybridized carbon atom
in its molecular surroundings; it is the negative of the
corresponding atomic valence state ionization potential. 13 is
a measure for the interaction between two such electrons on
different carbon atoms, 1.34 l apart. It may be calibrated
empirically:
Thermochemical calibration of 13:
i) For test molecules the enthalpy of formation AH is d~duced
from measured heats of combustion. It is compared with AH
computed from additive increments for molecular fragments.
The difference is attributed to a resonance energy (see
page 12). The result is 13 - 15 to 20 kcal/mol.
ii) The barrier to internal rotation in ethylene, which is about
25 kcal/mol, is set equal to 213. The result is 13 - 12 to 13 kcal/mol.
9
Spectroscopic calibration of ~:
The longest-wavelength electronic transition in ethylene is
(in part) composed of the ~ ~ ~* transition. It occurs
roughly at 180 nm. A- 1 = 55'000 cm-1
hv = bE S'" 3.5 eV
We note:
Thermochemical predictions require thermochemical calibrations
of S on a test molecule;
spectroscopic predictions require spectroscopic calibrations.
Exercise:
Butadiene; butadiene with symmetry orbitals; analogy with
solutions of the free electron in a box.
Example: Benzene
6
~ ~ 0 0 © '05 2 4
3
Figure 2 Benzene, numbering of atoms, 2p~ orbitals
1 2 3 4 5 6 We divide each
1 a-€ ~ 0 0 0 ~ column by S and
2 ~ a-€ ~ 0 0 0 abbreviate
3 0 S a-€ ~ 0 0 0 £...:.....£ = x
4 0 0 ~ a-€ ~ 0 S
5 0 0 0 ~ ~ We thus get the secular a-€ equation in the form:
6 ~ 0 0 0 ~ a-€
10
x 1 0 0 0 1
1 x 1 0 0 0 One obtains the
solutions: 0 1 x 1 0 0 2k71 0 xk - 2cos b' where 0 0 1 x 1 0
0 0 0 1 1 k 0, ±. 1, ±. 2, 3 x
1 0 0 0 1 x
We then obtain the following energy level scheme and eigen-
functions
3 e3 a -
+2 -2 e±2 a -
-------------+1 n T! -1 et1 a +
H 0 eo a +
~ form of the molecular orbitals:
!J'+2
213
13 tJl+?, CP-2
13 CP+1 CP-1
213 CPo
01
- :: -/ \
/
-$- ~\ ;' l \
/ \ \
6 -$--I
(1 Figure 3 Real benzene MO's
11
Complex rorm or the molecular orbitals:
b k f (-It 'Xp = k L exp (+3p'2'1fi/6) • 'Xp p=l P
N
e:!:2 .1.. L exp (±. 2p·271i/6) • 'Xp These complex orbitals ='{6
p=l are symmetry orbitals
or the subgroup C6 or 1 L exp (±. p'271i/6) . 'Xp D6h. e"!:.1 ='16
p
a .1.. L 'Xp CPo V6 p
The relations between the degenerate real and complex solutions
are
CP+1 ;
1l'-1 ;
Later on we will see that it is more convenient to use complex
orbitals than the real ones. Physically they are or course
equivalent, for any linear combination of two eigenrunctions
belonging to the same eigenvalue is again an eigenrunction to
that eigenvalue.
3. Some important derinitions N
Atomic density:
(Total atomic population)
L bicrrcir i=l
; { occupation number bi = 0, 1, 2
It is a measure of the amount of 'If electrons on atom p. For
all a1ternant hydrocarbons in their ground state
~ = 1 on all atoms.
Bond order:
12
N
I biC~rCiS i=l
0, 1, 2
Within the frame of RUckel theory this is the first-order
density matrix.
Energy of a configuration: It is the sum of the one-electron
energies of the TI electrons in the system of interest. N
E I b i €i
i=l
The lowest configuration possible is the ground configuration.
In it the lowest one-electron levels are all doubly occupied.
It is an approximation to the many-electron ground state. For
benzene we have
6a + 813
Resonance energy: It corresponds to the difference
{EG (TI electrons completely delocalized) -
EG (TI electrons localized in double bonds)}
For benzene this }
may be visualized as l ©] [0] and amounts to
EResonance (6a + 8(3) - (6a + 6(3) 213
Alternant hydrocarbons: They may be divided into alternate
nonneighboring st~d and unst~d carbon atoms
* * * * * ~
t *~* v.v
* 1 *
* *
13
In nonalternant hydrocarbons there
occurs at least one bond between
two starred (or two unstarred) atoms.
In alternant hydrocarbons energy levels are pairwise equidistantly
spaced with respect to a (provided overlap is neglected):
E:i a+xi 13 CDi = I*ciPXp + IO c. ,X I J.p P
p p'
E:(N+l-i) a-xi 13 t'lJ(N+l-i) = I* ci X ~ P P
_ I O ciplXpl
p p'
The corresponding eigenfunctions are characterized by the
fact that for starred atoms the coefficients are the same
and for unstarred atoms they are of equal absolute value,
but opposite sign. The absolute designation of an atom as
"starred" or "unstarred" is of course arbitrary, but such
is also the absolute sign of the molecular orbital.
The so-called "pairing" of electronic states in alternant
hydrocarbons has its physical consequences. For instance, it
follows from the HUckel model that in the radical anion and
radical cation of an alternant hydrocarbon the spin distri
bution should be identical. The experimental proton hyperfine
splittings in the electron spin resonance spectrum of such an
anion and cation are indeed remarkably similar [4], provided
they are unsubstituted.
The inclusion of heteroatoms:
Much has been written and said on this topic.
In general one may write:
14
Examples [3c] 0 0 Oc=o 0 H
h' N 0.5 - 1.0 hN 1.5 h' 0 1. h" 0
kCN 1. kCN 0.8 kCO 1. kCO
In general one may assume ~Cx ~ ~ . scx, ~ being a proportion
ality constant. Caution must be exercised in applying this
relation to 3d.-row elements.
Some useful relations:
From E = I biei follows (without proof)
i
1 bE '2 b~sr
Furthermore one defines:
bQs = rrS,q atom-atom polarizability
baq
bPsr = rr bond-bond polarizability b~qt sr,qt
These quantities prove useful in applying perturbation theory
to the HUckel procedure, as they are related to the first and
higher derivatives of the energy with respect to basic
quantities, and as they may be computed from eigenvalues ei
and eigenvectors cip' The pertinent formulae are to be found
in ref. 3f, for instance, and both atom-atom and bond-bond
polarizabilities are tabulated for a large variety of hydro
carbons in ref. 3e. However, due to increased computer
facilities, the use of perturbation techniques in the frame
of HUckel theory has lost some importance in recent years.
2.
0.8
15
III. Many-electron theory of ~-electrons
We now explicitly consider the interaction between ~-electrons
(5J. Our Hamiltonian has the form
and we no longer consider each ~ electron to merely move in
the inaccurately defined average field of all the others, but
rather to depend more explicitly on their relative positions.
We make the following formal assumptions:
a) Each electron U may in zeroth order of approximation be
described by a spatial one-electron function or orbital
(in our case a molecular orbital, MO) ~(~).
b) Each electron is, with respect to all the others, in a
definite spin-state a(u) or ~(u). We thus associate with
every electron a spinorbital ~(u)'a(u) or ~(u)·~(u). In
the following sections we will often abbreviate ~ • ~,
~ =~.
c) Any many-electron function must be antisymmetric with
respect to the exchange of two electrons, as required
by the Pauli-principle.
Consequently such a function is best represented by a
Slater determinant or by a linear combination of Slater
determinants, each such Slater determinant being an anti
symmetrized product of one-electron functions, i.e. of
spinorbitals.
1. Ethylene as two-electron problem
We denote the two carbon atoms by a and b and invoke the
cr-~ separability. We thus have a pseudo-two-electron problem.
Starting from our previous one-electron energy level scheme,
16
we can construct 6 configurations consistent with the Pauli-principle.
---1{J2 1 !
H I{J, t t 'Pc.= CPo (t)CP2
1 (2)CP2
1
Of course: 1 Cj)1
V2(1+Sab)'
CP2 1
V2(1-Sab)'
and the ground configuration
1 ~G = V2'
CP1 a ( 1) CP1 t3 ( 1) I CP1 a (2) CP1t3(2)
a b
~ Figure 4
1
! ! (3)CP2
1 (4)CP2
1 cp22
11
Figure 5
{Xa + Xb}
{Xa - Xb}
is wri tten:
v?{CP1a(1)C01t3(2) - Cl'1t3(1)CP1 a (2)}
~~1 (1)7'1 (2) {a(1)t3(2) - t3(1)a(2)}
We abbreviate ~G:; iCP1 CP1 1 , namely as the diagonal part of the Slater
determinant, omitting the normalization factor. The excited
configurations are consequently written:
17
All the singly excited functions are degenerate to zeroth
order.
l2.)~
(.31~~
(4)~~
IC01 q)21
1q)1 q):d
1Ci>1 q)21
ICP1CP21
l2)E~ C~)E~ (4)E~ = a + 13 + a - 13
(in the HUckel approximation)
We assume that a true description of the electrons which we
consider will not be given by ~ configurational function,
but by a linear combination of them:
To determine the correct expansion coefficients AA we apply
the variational principle in formally exactly the same way
as we did in the one-electron case. This leads to a secular
equation
IX~AI - E SAAI I 0
where the indices A and AI run over all configurations. The
dimension of our secular determinant is equal to the number
of configurations of interest.
Now, before we compute the matrix elements of X~ in the
basis of the configurational functions, let us, however,
2a
18
consider the rollowing:
As the operators S2 and Sz ror the total spin commute with
the Hamiltonian (in absence or spin-orbit errects)
o o
matrix elements or KTI will vanish between conrigurational
runctions which are eigenrunctions or S2 and Sz with
dirrerent eigenvalues [6J.
-+ -+ -+ or course S S1 + s2 and Sz s1z + s2z
S2 2 2 -+ -+ s1 + s2 + 2s1 • s2
2 2 + - - + + 2s1 z • s2Z s1 + s2 + S1 • s2 + s1 • s2
where + -s1 = s1 X + iS1 Y , s2 E s2X iS2y etc.
s1Za (1) ~ a(l) S1zfl (1) h We remember that - - fl(l) 2
+ a(l) 0 + fl(l) h a(l) s1 s1
s1 a(l) h fl(l) 51 fl(l) 0
2 a(l) = ih2a(1), 2 fl(l) i h2fl(1) s1 s1
etc.,
consequently:
Sz 4>G 0 and S2 4>G 0
S (1)4> 2 Z 1
l.h. (1)4>2 1
S2 (1).p2 1
2·h2 .(1).p2 1
S (2).p2 z 1 0 S2 (2l.p2
1 h2f2).p~ P).p; }
S (3).p 2 Z 1 0 S2 (3).p 2
1 h2{(2).p~ +(3).p; }
S (4).p2 Z 1
_1·h.(4).p2 1
S2 (4l.p 2 1
2 .h2. (4).p2 1
Sz .p22
11 0 S2 .p22 11 0
19
Exercise: 1) Derive the above relations
Our above functions are already eigenfunctions of SZ. We are
now in a position to write eigenfunctions of both Sz and S2. (2S+l) j We designate these functions by Ms ~i ' where Sand Ms
are total spin and z-component-of-spin quantum numbers.
Singlets Triplets
!~G I q>,. (PI I 3~'l -1 1 1~1cp2\
:~~ = ~ {1~1cp21+1~2~1 I} 3~2 o f = k {1~1qs21-1~2qs11}
~~2,2, o 11 \~2;P21 3~'1
1 1 = \~1 ~21
We now make use of these 6 functions to describe the electronic
states of ethylene. As mentioned, matrix elements of Xn
will vanish between functions of different S or Ms values. So
the triplet functions, as they stand, are already solutions.
We now compute nonvanishing matrix elements of
n e 2 X = hcore(l) + hcore (2) + --
r12
between the Singlet functions. The matrix elements are ex
panded into one and two-electron integrals, as indicated. Only
the integration over the spatial variables is explicitly
mentioned. Of course, the orthogonality of a(~) und ~(~) must
also be observed. hcore(~) is just written h(~), where
\..l = 1,2.
+
S\ql1~ \*h(l) \ql1~ \dT1dT2 + id. for h(2) +
SI~1CP1\* ~2 \tP1CP1 IdT1dT 2
Upon expansion of the determinants:
20
For this expression we use the abbreviations
or equivalently
= 2h11 + (11111)
Similarly:
(~~Glx71I~~~) = &Jltp1~11*h(1) {1C(l1~al+I(I)2cp11} d'l"1d'l"2 + id. h(2)
+ k J 1(1)1 q;1 1* ~: 2 {I tp1 q;21+ 1q>2CP1 I} d'l"1 d'l"2
= '12' {(q>1l h lq>2) + (q>1ql11<:p1~2)}
= Y2 {h12 + (11112)} 0
(To ascertain this result, the integrals over molecular orbitals
must be expanded into integrals over atomic orbitals~ see below.)
In the same way we find:
( 1 ~ Ix7111 ~'I.'I.) o G 0 14
(~~~ Ix711 !~~)
(~~~ I X711 ~~!~)
(~~~~ IX711 ~~~~)
(11122)
h~ + h2.~ + (12112) + (12121)
V2' {h •• + (22121)} = 0 to ascertain this, .. ~ see below)
2~2.. + (22122)
From these results the singlet secular equation is obtained.
21
1 1 2.2. 1 ::1 atG at "1 at 1 -----------------------------r-----------------------1
2h1f+ (11\11 )-E (11\22)! 0 I I
(11\22) 2hu+ (22122)-E ! 0 1 -----------------------------:-----------------------
o 0 :hlf+hu+(12Il2)+(1212l)-E -----------------------------~------------------------
We notice that the equation factorizes into a (2~2) equation,
connecting the ground configuration with the doubly excited
configuration,and into a (lxl) equation, of which ~t; is an
eigenfunction with eigenvalue
Consequently, to summarize, our singlet eigenfunctions
have the form:
1 lira
1t2 o t
The coefficients AI and AII have to be numerically determined,
but evidently AI » AII •
The triplet secular equation has no off-diagonal elements
for reasons already discussed. Indeed we have
h11 + h'L'l + (12 112) - (12121)
The three triplet eigenfunctions are given on page 19.
Figure 6 shows at left the relative energies of the various
configurations, neglecting electron interaction, and just
o
22
summing the one-electron energies. Note the degeneracy between
1~~ and 3~~. At right, in a somewhat arbitrary scale, are the
results with electron interaction. A striking feature is the
energy splitting between singlet and triplet .1' no electron interaction c - c;.r with electron interaction
,,,..22 .,.,,,-3cpf
'~-
r~---~A~------~, }2(12 12 T> _
'ep -G
The use of spatial symmetry:
Figure 6
~~ ___ ~y~ ______ ~J
31/1,
(The reader is here assumed to be familiar with elementary
group theory)
The molecule ethylene has the symmetry D2h ,
The MO CP1 transforms according to the irreducible representation blu'
The MO CPa transforms according to the irreducible representation b2g•
(We use small lettering t·o characterize one-electron states and
capital letters to label many-electron states,)
Consequently ~G transforms like Ag blu ® blu ~'l. " " B3u blu ® b2g 1
~·I.'L " i1 " Ag b2g ® b2g
23
As XW transforms like Ag, matrix elements between configura
tional functions belonging to different irreducible represen
tations vanish. This explains the factoring of the singlet
secular equation.
The evaluation of integrals:
For numerical computations integrals over molecular orbitals must of course be further expanded into integrals over atomic
orbitals. In this sense we obtain [5J:
a) Two-electron integrals:
(1)(11 (1l(t)
<aalaa> + <ablab> + 2<aalbb> + 4<aalab>
2(1+Sab)2 (1)00 (11l7.J
<aalaa> + <ablab> + 2<aalbb> - 4<aalab>
2(1-Sab)2
(121 12) = J ,'2 ~aalaa~ + ~ablab~ - 2~aalbb~
2(1-S2 ) ab
(12121) = K.'2 ~aalaa~ - ~ab lab~
2(1-sib)
Consistent with our previous notation
S \.l v (ablab) = Xa (1)Xb (2)
(aalbb) = j'Xa (1)xa(2)
(= (aalbb))
(= (ablab»)
The round-bracket notation is used in [5J. These integrals
are either calculated accurately or evaluated semiempirica1ly.
The semiempirical evaluation is important in our context,
i.e. in large molecules, and will therefore be dealt with:
- Zero differential overlap approximation: Wherever the
product Xa(l)Xb(l) occurs it is neglected,unless Xa = Xb·
With this approximation Sab = 0; (aalbb) = 0; (aalab) = O.
24
- We approximate one-center two-electron integrals as
(aalaa) ~ Ia - EAa
where Ia is the potential of ionization of an electron
located in the 2p~ orbital Xa on the Sp2 hybridized carbon
atom a in its molecular surroundings, and EAa is the
corresponding valence state electron affinity. The above equation may be visualized as the transfer of an electron
from one carbon atom in the given valence state to another identical one infinitely far away. The energy required to
carry out that transfer is, on one hand, Ia-EAa, on the other it may be viewed simply as the work required to over
come the repulsion energy (aalaa).
--------------~~
+ Figure 7
Two-center two-electron integrals of the type (ablab) may be semiempirically approximated by an electrostatic model (for
details, see next section).
b) One-electron core integrals:
<alhla> + <blhlb> + <alhlb> + <blhla> 2(I+Sab)
<alhla> + <alhlb> I+Sab
25
We consider the different terms contained in the core operator
h(l) :: h(Il):
h(l) = T(l) + Ua(l) + Ub(l) + '-----y-----' InI:eractim of the electron w:iih the positive oore at:' cmbon atoms a and b
The matrix element of the
sum of the first two terms
between identical functions
Xa may be viewed as the
negative of the "valence
state" ionization potential
{UH1 (1) + UH2 (1) + UH3 (1) + UH 4 (1)} , ~ .
Relatively small interaction of the electron with the neutral hydrogen atoms. These terms are in general considered negligible
Figure 8
The matrix element of Ub between identical functions Xa may
be considered as the interaction of an electron in Xa with
the positive hole created by the vacancy of an electron in
Xb' and set approximately equal to:
,.., - (ab\ab)
Thus:
The matrix element of h between orbitals on different centers,
Xa and Xb' is difficult to interpret physically term-by-term.
For larger ~ electron systems (see also section III.3) the
core resonance integrals
26
l3ab <a\h\b)
are generally calibrated on a test molecule. The quantities
aa and l3ab, as described here, are not to be confused with the more crudely defined coulomb and resonance integrals of
HUckel theory, in spite of the similarity.
2. The configuration interaction (eI) procedure
The procedure outlined here is, in principle, applicable
to any many-electron system. Our formulation is consequently
not limited to n electrons.
We consider a molecular system with 2N electrons, described
by a Hamiltonian
I h(~) + I I v(~,v) ~ ~>v
e 2 where v(u,v) = --
- r~v
The wavefunctions for the many-electron system will be linear
combinations of Slater determinants, which in turn are
defined as antisymmetrized products of spinorbitals (spin
MO's). The MO's are written as linear combinations of basis
orbitals (AO's):
p
We assume that the MO's are ordered with respect to an energy
criterion. They are, in the case of ~ electrons, for instance,
HUckel MO's ordered in the sequence of increasing energy. We
are thus able to define a
ground configuration ~G
singly excited configurations,
27
~~~ = 11:1>1 rp1 ••• rpi epk ••• I:I>i'N1
triplet !~~ = A {II:I>1«>1'" qJi~k'" Cl'lfPNI-1 1:I>1(P1
~~~ 11:1>1 rp1 .•. !:Pi Cflk ••• ~«>N I
doubly excited ~~~, triply excited configurations ~ki~n , etc. ~J Jm
(see Figure 9). In the following we restrict our computations
for simplicity to the ground and singly excited configurations.
The general solutions to the problem
X, E. will accordingly have the form
000 un
'4rG (o)A ~
G G +L L(o)A~ ~k i + ••• '0 ; Eo
i k roc un
{:n= ·n (n)A ~ +L L (n)A~ ~k + ••• *n 1,2 G G i i k
The designations "occupied" orbitals i and "unoccupied"
orbitals k refer to the ground configuration (see Figure 9). The problem now consists in finding the eigenvalues En and expansion coefficients (n)A (n = 0,1,2 ••• ). This is done by
diagona1izing the matrix of X, i.e., in solving the secular
equation, in the basis of the configurational functions ~G k
~i' etc. (see also section III.1). This procedure is called
configuration interaction and is a very general method for treating many-electron problems. If the configurational
functions are in any way reasonably conceived, the coefficient
{o)AG should be large (see also section IV.1 and IV.3) and the coefficients (I1)AG should be small.
28
CPo cp;k cpkl r • 'j
unoccupied { 'PI I I . t I
(virtual) 'Pk k t k t k
orbitals 'PN+1 .
N+1 N+1 N+1
ffJN 1,! N H N H N
occupied ffJJ --4-j H j l j
orbitals ffJ, H i 1 ; ! ; . . . . .
ffJ, H 1 H U 1
Figure 2
We now turn to the necessary evaluation of matrix elements:
To this end we make use of the Slater-Condon rules for matrix
elements between Slater determinants r6]. We find
N
(~GI~I~G) = I 2(~ilhl~i) i=l N N
+ I I r2(~(u)~j(v)lvl~i(U)~j(v» i=1 j=l
- (~i(U)~j(V) Ivl~j(U)~i{V»J
li! I 2(i \h\i) i
+ II {2(ijlij) - (ijlji)} i j
5 I 2hii + I I (2Jij - Kij ) i i j
o \1
til
\1
29
Integrals of the type Jij are called coulomb integrals,
integrals of the type Kij are designated exchange integrals.
The double summations are here taken independently over
spatial orbitals.
We now abbreviate N
Fik = (ilh\k) + I {2(ijlkj) ~ (ijljk)} and find [7]: j=l
(9G \JC 11<i~)
t9~IX11 9~)
(1tilxI19~)
(1 <i~IXI1 t~)
(l<i~IXI f~~)
(3~~IXll~~)
(l~~IXI3~~)
~Fik
(~GIXI~G) - Fii + Fkk - (iklik) + 2(iklki)
- Fmi - (mk\ik) + 2(mklki)
Fkn - (iklin) + 2(ik\ni)
- (mklin) + 2(mklni)
(~GIXI~G) - Fii + Fkk - (iklik)
- Fmi - (mk\ik)
Fkn - (ik lin)
These formulae are exact.
Exercise a) Verify the above expressions for the two-electron
case and compare with section III.l.
b) Verify for the case of the 6-electron problem
III 22 331.
30
3. The semiempirical PPP approximation for ~ electrons [8]
The matrix elements between configurational functions obtained in
section III.2 lead to the evaluation of one- and two-electron in
tegrals over one-electron functions (MO's). Here we consider the
semiempirical evaluation of these integrals in the frame of ~ electron theory. Consequently we treat in a more general fashion
some points already mentioned in section 111.1., p. 23-26.
a) Two-electron integrals: We write
(Il)
(ijjkt) '-----'
(\I)
and expand
CPi I CipXp P
CD j I CjqXq q
CPk I CkrXr IIII * * CipCjqCkrCts (pqlrs) r p q r s
CPe I CtsXs s
(pqlrs) = J X~(1)X~(2) ;2 Xr(l)Xs(2)dT1dT2 12
= (prlqs) in Pariser-Parr notation [8]
The functions Xp,Xq here. of course, represent 2p~ AO'S on atoms p,q respectively.
Neglect of differential overlap (ZDO-approximation):
leads to the simplification
(pqlpq)
I I c~PcjqCkPCeq (pqlpq) p q
(pplqq) is also often designated ~pq.
31
An estimate of Ypp (see also 11.1, p. 24) is given by the relation
y ""I -EA pp P p
where Ip' EAp are the valence state ionization potential and
electron affinity of atom p, respectively.
For Ypq (p ,; q) Pariser and Parr suggest the "uniformly
charged sphere" approximation (see Figure 10):
Charge density
Slater effective nuclear charge
Figure 10
The radius Rp is given by
R = ~ X 10-8 cm p Zp
R
The number 4.597 is obtained by setting the "electrostatic"
value of Ypp equal to the analytical value.
The simplified form of the electrostatic formula, valid for
R ~ 2.80 A reads:
2 -1/2 R +R 2 -1/2
Ypq = 7.1f75 {[ 1 + (Rp2-:q) ] + [1 + (~) ] } eVe
(For R < 2.80 A, see [8J.)
32
Figure 11 shows the value of Ypq where p and q refer to C atoms as a function of distance
R = 0
~~
" • ~ \. ~ '. ~.-
R~ ~ ~
Fisure 11
b) Core intesrals: The one-electron part of the Hamiltonian,
h(~), refers to the kinetic energy and the core potential. Within the PPP approximation the core consists of the molecule,
fixed in its equilibrium geometry, minu& the ~ electrons. Each
atom contributing one/two ~ electrons carries a positive
charge of one/two in the core. The other atoms of the molecule
(such as H atoms) also are part of the core, but are assumed
neutral (see also III.l, page 25). Upon expansion
(ilhlk) = I I c~pckr (plhlr) p r
we now distinguish different cases
{ p = r, hpp = up ; see below
(plhlr) S hpr p and r are nearest neighbors,
p and r are not n.n., hpr = 0
~pr is best calibrated on the spectroscopic properties of a
test-molecule. It is generally of the order 2-3 eV.
As already stated above and from the point of view of atom p
we subdivide:
hcore(u) T(u)
Kinetic term
+
33
:Ji1ta'1:ction with Intera.cti.on w. lhterootion with atomic cxre p o1her atomic neutral atoms atta:med
(Dres q;ip to the core, sooh as H-atoms
As we have seen previously:
Ip is the "valence state" ionization potential for an electron
in the 2p~ orbital Xp of the sp2-hybridized atom p in the
molecule.
Furthermore:
Uq(U) = U~(u) - z~) J X~(V)Xq(V) ~:V dv
i.e., the interaction with the core atom q is equal to the
interaction with the neutral atom q, Uq(u), minus the inter
action with the "missing" 71 electrons on that atom, the number
of which being z~). For carbon J71) = 1, for nitrogen i 71)= 1 in
pyridine, but J: = 2 in pyrrhOle; for instance. q
Consequently:
Up = -Ip - I {z~) (pqlpq) - (pIU~IP>} + I (pIU;lp) q;ip r
Neglecting penetration integrals, i.e., integrals corresponding
to the interaction with neutral atoms, or incorporating these
terms implicitly into an effective valence state ionization , potential Ip' the formula simplifies to the "working" ex-
pression
i 71) • 'Y q pq
Let us illustrate this by some examples:
34
-f2 - 'Y12. - 'Y13 - 'Y14 - 'Y15" - 'Y1'
Benzene 1(1) 2 2 - C - 'Y12. - 'Y13 - 'Y14 ,
{ (2)
a., -IN - 'Y12 - 'Y13 -'Y14 - 'Y15
Pyrrhole (2.)
2'Y12. - 2'Y13 • but -IN -
(12 = _f,1) _
2'Y21 - 'Y23 - 'Y24- 'Y25 C
The nitrogen atom in pyrrhole contributes two ~ electrons, 1'1.) thus IN refers to a valence state ionization potential for
double ionization. In the expression for (12' 'Y21 must (~) accordingly be counted twice, as Z1 = 2.
H H H
H H H:O:H + + +
+ + +
H H H N H H H
H
Figure 12, PPP cores
4. Benzene as an example
The ground configuration
of the ~ electron system
of benzene
is represented in Figure 13.
35
b
e+2
e +1 H a
e-2
H e-1
it Figure l~
To discuss the lowest excited states we limit ourselves to con
sidering only the four degenerate singly excited (singlet)
configurations (see also Section II.2):
The one-electron functions (MO's) in their complex form are
e j =kIw jp Xp j 0, ±.l, ±.2, 3
p
and UJ ;: e 2~i/6
Corresponding to their transformation proper-
ties under the point group Cs we write eo ;: a and e:3 0; b.
Including the ground configuration, but neglecting higher
configurations, the problem would lead to a 5 ><.5 configuration
interaction secular determinant. Full exploitation of symmetry
simplifies the problem decisively, however. In the Dsh point
symmetry of the molecule the ground configuration only mixes
with doubly excited configurations and ~o thus is a relatively
acceptable description of the ground state. (See also Section
IV.3 on the Brillouin theorem.)
36
*0 ~ ~o' transforms like A1g under Deh· From the way the oneelectron functions transform under Dsh, we may also deduce how
the configurational functions transform [9]:
.1 1 {ill d2 ~-1 -2 }
- ~+1 transforms like B:m
'2 L {~+2. V2 -1
-2. } + ~+f transforms like B1U
1 {-t _ ~+'2.} } .3 iV'2 ~-1 +1 transform like E1U
1 {-2- ~+2.} and must be degenerate
*4 V2 ~-1 + ~ I
There are no matrix elements between functions transforming
according to different llTeducible representations of the same
group. Consequentl~ the excitation energies are directly
given by the diagonal matrix elements between the above
functions. Taking this into accoun~ and upon expansion:
E1(1B2U) ('1 13(7f I .1 > Eo + h22 - h11 + K02 - K03
E2(1B1U) (t2Ilc7f lh> Eo + h22 - h11 - K02+3Ko3
E3CE1U) (.3I lC7f lt3> Eo + h:?2 - h11 + 2K01 - K03 4 4
where, of course: e 6 6
3(7f I h(\.l) + I I L \l=l \l>v=l
r\lV
Concerning the evaluation of the above matrix elements [5]
A two-electron integral is written
(eieklejee> 5 (iklje> is equal to the expansion
and with the ZDO approximation,
37
(ik I H) 1 \\ (j-i)p+(e-k)q
3b L L llJ • ~pq P q
It may be shown that due to the cyclic symmetry this integral
vanishes, unless
(j-i) (k-e), or (j-i) (k-e) ± 6n, where n = 1,2, •••
All Coulomb integrals (ik\ik) are equal, and the exchange
integrals (ik\ki) reduce to three types, namely K01 , K02 and
K03 , where, for instance:
i k j e (O±ll±l 0)
(±1±2\±2±1)
K01 } = * {'Y11 + 'Y1 2 - 'Y1 3 - 'Y14}
K01
and less obviously:
(-2-1\+2+1) modulo 6 K02 * {'Y11 - 'Y1 2 - 'Y1 3 + 'Y1 4}
(-2-1\+1+2) modulo 6 K03 * {'Y11 - 2'Y1 2 + 2'Y1:3 - 'Y1 4}
In conclusion we obtain the following energy level diagram
(Figure 14).
'E,U ZOeV ",,-2 "'+2 '+'-1 , .... '+'+1 - - - - - 'B,u B.2eV - 'f - I 'B2u -4.geV
If I I I
I I I I I I I L ~'9
Figure 14
38
Listed are the experimental energies above the ground state.
For a comment on the transition probabilities, see the next
section.
Exercise: Verify the energy expression for the 1B2U state.
5. Electric-dipole transition probability
The semiclassical Hamiltonian for a many-electron system in
a radiation field (within the B.O.-approximation) is written
[2]
X = )'.L {p _ ~ Jt (t)}2 + V tr2m \-l c\-l
where t\-l(t) is the vector potential of the field at the site
of electron \-l at time t. V contains all electrostatic terms.
e designates the (negative) charge of the electron.
V I U(\-l) + I I r:: \-l IJ>V
X can be separated into a time-independent part Xo and a time
dependent part X'(t) containing the perturbation due to the
external field. It can be shown that the time-dependend part
may be written as a mu1tipo1e expansion, convergent if the wavelength of light is much larger than the dimensions of the
molecule (A > 1000 1; L - 10-50 1) [10]:
X' (t) = -E(t).it - H(t).M + .... .. .... E(t) and H(t) are respectively the electric and magnetic
radiation field at some chosen pOint in the molecule. The
probability for a transition from a (ground) state .a to an
excited state 'b is by time-dependent perturbation theory [2] proportional to:
39
In this expression the contribution of the electric dipole
term turns out to be by far the most important one. We consequently write. after averaging over randomly oriented molecules:
... Dab is called the dipole strength and R is the electric dipole
operator: ReI e;u . U
The integral <'aIR1'b> is called the transition moment.
For a many-electron (2N) system we find:
and
lib ,.. Bo9?o + I I B~9?~ + i k
where as usual
Consequently
Bo<9?ol~l9?o> + II B~(9?ol~I9?~> • i k
... As Bo is in general small, the first term (where (9?oIRI9?o> within our approximations is the expression for the dipole
moment of the ground state) may be neglected.
Thus to a good approximation
40
'@ e I I B~<CIli \;\CIlk) i k
The problem then boils down to the evaluation of matrix
elements of the operator ~ between one-electron MOls.
I I C~p Ckq <'X.p\~\'X.q) P q
In the frame of PPP calculations on ~ electrons the following
procedure is admissible to estimate orders of magnitude:
.. rop is the position vector of atom p with respect to the
origin of the coordinate system. This expression is exact. The
next expression is only exact between like atoms and orbitals. .. .. .. (rOp+r09) <'X.p\r\'X.q> ~ Spq 2
If one is schematically consistent in neglecting differential
overlap, these cross-terms may even be neglected.
Some symmetry considerations:
A transition a~ is called electric dipole allowed if the
integral <'a\R\'b) = S ': R tb dT fails to vanish. Not to vanish, this integral must transform according to the
totally symmetric irreducible representation of the point
group of the molecule. On the other hand, the integral .. <'a\R\'b) transforms like the triple direct product of the
irreducible representations according to which 'a' Rand
Vb transform respectively. Thus ra®rit®rb must contain the
totally symmetric irreducible representation.
To revert to benzene as an example: ra = A1g, so rit®rb
must contain A1g for the triple product to contain A1g.
This can only be the case if rit = r b • The electric dipole
41
operator R transforms like the vector components x,y,z. The
components x,y in the plane of the molecule transform like
E1U. So rb must be identical with E1U for a transition to
be electric dipole allowed in the plane of the molecule. We
then summarize
Transition A1g ~ E1U electric dipole allowed in the plane of the molecule
Transitions A1g ~ B2U } electric dipole forbidden A1g ~ B1U
If the latter transitions still appear in the spectrum,
this is due to vibronic coupling with normal modes of
appropriate symmetry.
42
IV. Self-consistent-field (SCF)-methods
SCF calculations essentially follow the method proposed by
Hartree and Fock about 40 years ago and applied by these
authors to atoms. It is a variational procedure (analogous
to the simple Ritz method, see section 11.1., page 5) taking
to some extent electron interaction into account explicitly
(l1J.
1. Simple LCAO-formulation of the closed-shell case
We consider a 2N electron system and assume that it can be described by an antisymmetrized function
The Hamiltonian be: 2N 2N
I h(ll) + I
We now seek a set of one electron functions ~1 ••• ~i ••• ~ . such that
J ~~modT J ~:~odT be a minimum.
From the variational principle we may assume that
the minimized energy will approximate the true ground state energy.
We must, of course, impose the constraint that the set of ~i be orthonormal, that is
for all i,j
43
Assuming ~o to be normalized, we find (as on page 28):
Coulomb-term N N N
Exchange-term
L 2(Wi \h\Wi> + I I {2(WiWjIV\~i~j> - (~i~jlvl~j~i>} i=l i=l j=l
The double summations go independently over spatial orbitals.
With the LCAO-expansion:
M
Wi I CipXp p=l
M
~j I CjqXq q=l
M
* I * * ali cirXr r
M
* \' * * ~j L cjsXs
s
We may write:
~rspq
II r p
r s p q
IIII r s q p
[1) (ll (1) (2)
<XrXs \XpXq >
M: number of basis functions
* * CirCjsCipCjq~rspq
* * CirCjsCjqCip~rsqp
(I) (i) (1) C2.)
<XrXs \XqXp>
44
Introducing these expressions into the one for Eo' we see
that Eo becomes a function of the coefficients:
Eo (cip; i=l .•• N, p=l .•• M)
Eo (c~r; i=l .•. N, r=l •.. M).
or equivalently
To minimize this function under the constraints
= I I r q
we obtain the equation below.
'----...... v------'
takes:hID account N orthon:rmali t.v constraints
o r 1 •••• M
Aij is a Lagrangian multiplier. It may be shown (see section
IV.2) that one may set
thereby simplifying our equation to
o
For computational reasons we assume that we consider a par
ticular value of the indices i and r.
To remember that we consider particular values of i and r let
us set i = i', r=r' and evaluate the above expression. In
differentiating the different terms of Eo we must exercise
some caution. For instance, in the exchange part
* the term in ci'r' will occur twice, once for i=i', r=r', and
once for j=i', s=r', leading to the respective derivatives
45
and
It is easily seen that both terms are equal, leading thus
to a factor of 2 in the general expression. A similar
situation is of course encountered in the coulomb part.
We thus obtain:
2 I ci'phr,p + 4 I Ci'p I I C;sCjq~r'spq p p j sq
Dividing by two and abbreviating N
I C;SC jq == j=l
(==
we find:
I Ci'p {hr,p + 2 I Dsq~r'spq - I Dsq~r'sqp - sr'pe1,} 0 p sq sq
The index i' is now redundant and {P = 1 ... Ml We thus have r' =r=l ••• Mf·
M linear homogeneous equations with M unknowns. The existence of nontrivial solutions requires the determinant of the
coefficients to vanish:
detlhrp + 2 L Dsq~rspq - I Dsq~rsqp - Srpel 0 B:! sq
I
46
p = 1 ... M, r = 1 ... M. The first three terms are customarily
designated by Frp and are the matrix element of the ~
operator F in the basis Xr ' Xp' To solve the problem one
must:
1. Choose an appropriate basis Xp' The bigger M, the better.
M should always be significantly greater than N. For
M=N no energy lowering will be achieved. In this case the
energy will remain constant, as (~oIXI~o) remains in
variant under a unitary transformation of the ~1 "'~N
among themselves.
2. Compute integrals ~rspq' ~rsqp' hrp ' Srp' However, the factors Dsq couple the equations together and require a
knowledge of the solutions ~i = ; cipXp ' One therefore
proceeds as follows:
3· Guess an approximate form for the CPi and compute approximate
Dsq'
4. Solve the secular problem a first time.
5· Recompute the Dsq with the new eigenvectors.
6. Solve the secular problem a second time.
7. Repeat the procedure until eigenvalues € and eigenvectors
converge (generally about 20 times). Then self-consistency
has been attained.
One thus obtains M SCF-eigenvalues: €1"'€M' corresponding
each to a respective one-electron SCF-function or SCF-MO:
~{C; ~~CF ••• tp~CF. Of these MO' s the N lowest ones are doubly
filled to approximate the many-electron ground state. The
M-N higher ones are virtual orbitals.
If the number of basis functions M becomes quasi-infinite, one
approaches the true SCF energy or Hartree-Fock limit. By
judicious choice of the basis functions one often succeeds in
ab initio calculations to come close to the Hartree-Fock
limit, even if M is finite.
47
2. Semiempirical simplification (ZDO-approximation)
Setting in all integrals X;(~)'Xq(~) 6 pq X; (~) • Xp (~) , the ~ are neglected except for r=p, rspq s=q, and the ~ rsqp are neglected except for q=r, s=p. Remembering that
for a particular matrix element rand p are fixed indices,
sand q running indices, the eigenvalue equation simplifies
to:
det\hrp + 20rp I Dss~rsrs - Dpr~rprp - brp €\ 0 s
Abbreviating ~rsrs !5! ~rs' ~rprp :; ~rp , one obtains:
+ 2 I Dpp~pp .} Dpr is here defined as
F h D ss~ps -pp pp I Cjp Cjr s j
F h - D pr~rp r + p For real orbi tals Dpr=Drp • rp rp
3. More general formulation of the closed-shell case
In the previous sections we optimized the one-electron
functions by varying LCAO-coefficients solely. While this is
very often done in practice, the Hartree-Fock problem may,
however, be more generally formulated without specifying how
the ~i are varied to minimize Eo' We start out from our
energy expression: N
E = Eo I 2<~ilhl~i) + i=l
N N
I I {2<~i~jlvl~i~j) i=lj=l
* and consider the functions ~i' ~j as independent variables.
To minimize this expression with respect to these variables
under imposition of the orthonormality constraints f .. :; lJ
<~ilroj) = 0ij' we may set the following total differential
equal to zero:
\' ~E. 6~i + \' bE * 6 * _ \' \' bf i.i ". 6 _ \' \' bf i.i ,,'" . & * 0 r u~l r bCPi ~i r ~-' b~i lj rt>i r ~bqJ~ iJ tpi
48
This is fulfilled if the factor of any b~i identically vanishes. We thus obtain a series of equations:
o • i 1 ••• N •
which leads to: (To verifY this following expression. write out E and f ij not in terms of brackets. but o£ integrals. Per£orm the differentiation within the integrals and set the sum of integrands equal to zero.)
{2h + 4 I < 'l' j I v I 'l' j >} 'l'i - 2 I < 'l' j I v I 'l'i> cP j - I q> /' j i 0 j j j
We also obtain a set of equivalent complex conjugate equations. 1 Dividing by 2. setting 2 Aji e €ji and abbreviating
•
We use the Coulomb operator J j and exchange operator Kj:
{h + I (2J j -Kj )} 'l'i j
It may now be shown
a) That the matrix of the €ji is Hermitian and may be
brought into diagonal form by a unitary transformation of
the CPi among themselves.
b) That such a unitary transformation leaves the Fock operator
operating on a function q>
{h + I (2J j -K j )} cP = Fer j
invariant.
We thus may assume a priori the CPi to be in the proper form
to write:
{h + I (2J j -Kj )} CPi j
!I
49
These pseudo-eigenvalue equations for the ~i are fulfilled
if the ~i are self-consistent. If not, we must solve the
equation F~ iteratively until the above relation is fulfilled
(see previous section).
From above it follows immediately that
&i' and <~iIFI~j) = &i(~il~j) (fer i+j)
o
This relation of course also holds for matrix elements of F
between filled and eventual virtual orbitals. We now
previously found (section 111.2, page 29):
(IPa 1:lC1IP~) = 'i2' Fik = V2' <~i lF1C11c)
It immediately follows that if the ~i' C11c are obtained by
the same SCF calculation, then these matrix elements vanish.
This situation is summarized as Brillouin's theorem: Matrix
elements between a closed-shell SCF ground state and singly
excited configurations defined within ~he same set of SCF
orbitals vanish.
(This theorem may also be stated in the reverse way. For
instance, we notice that the symmetry orbitals of benzene are
SCF orbitals within the ~ electron approximation, because
all matrix elements between the ground and singly excited con
figurations vanish.)
From
I (~ilh.+ I (2Jj-Kj)l~i) = I (~iIFI~i) i j i
and as
we immediately find:
N
I ( (i \ F \ i) + (i \ h \ i) )
i=l
50
N
I (€i+hii)
i=l
(This contrasts with the simple HUckel-type one-electron
approximation, where we had N
EHUckel '\ 2€ .• ) o L ~
i=l
The true SCF ground state energy is of course not the true
energy, as the interaction with doubly excited and other
configurations is left out. One generally defines within the
Born-Oppenheimer approximation
Exact nonrelativistic energy minus
Hartree-Fock energy = Correlation energy
4. Koopmans' "theorem"
Monoionization from a neutral closed-shell molecule may, to
an acceptable degree of approximation (with notable and
important exceptions), be pictured as the extraction of an
electron out of a given SCF orbital, the other electrons
remaining unaffected [llJ.
Suppose that the electron comes from a SCF orbital i (see
Figure 16), the energy of the ion will in that approximation
be given by (upper index: 1 singlet, 2 doublet, K Koopmans)
N
1E~CF - [(i\hli)+ I {2(ij\ij) - (ijlji)}]
j=l
Within this approximation it follows immediately that
2E~ =-1 ~on k k + i N + 1
51
1k being the ionization potential. Consequently
(-eN) gives an approximation of the ionization potential to the ground state of the monopositive ion, 11.
( ) { = 12, approximates the ionization potential -eN_l to the first excited state of the monopositive ion,
= 13, approximates the ionization potential to the second excited state, etc.
Koopmans' approximation neglects (Figure 17):
a) the reorganization energy of the electrons in the ion.
This reorganization energy is taken into account in the
restricted open-shell SCF energy of the ion (see Section
1V.6.1).
b) the difference between the electron correlation energy
of the neutral molecule and of the ion.
H eN 1 eN H eN
H H 1 eN-1
H H H
H H H H e, H e, H e,
'EsCF 0 'E sCF e o - N 'E sCF E: o - N-1
... 2E K
... 0 ... 2E K ... 1
Figure 16 Schematic representation of the Koopmans' energy of a monopositive ion 2EK
52
-T- -------I
2£:
R
~ -eN 2[SCF
a2 0
T 2[ 0
'E SCF 0
L1
1[ 0
a, ...L -----------
Figure 17 01: Correlation energy of the neutral closed-shell
molecule. R: Reorganization energy of the ground state of
the monopositive ion. 02: Correlation energy of the ground
state of the monopositive ion. ~: Exact ionization energy
to the ground state of the monopositive ion. Koopmans' "theorem"
states: -€N "" fj,. This implies: R+ 02 "" 01·
5. Some remarks on localized orbitals
The one-electron SCF orbitals ~i obtained by solving molecular
closed-shell Hartree-Fock-problems are called canonical
orbitals. They are the solutions of the pseudo-eigenvalue
equations F~i = €~i. Thus
~SCF o
S ~SCF X ~SCF dT o 0
SCF CPi = CPi
SCF Now it may be shown t~at ~o can be expressed in terms of
any set of orbitals ~i obtained by a unitary transformation
of the (doubly) filled canonical ~~CF among themselves, to
yield the same many-electron energy E~CF [llJ. These new , orbitals ~i of course no longer satisfy the pseudo-one-
electron equations.
It has for instance proven instructive to construct noncanonical orbitals according to certain physical criteria,
such as the criterion of maximization of electrostatic
orbital self-energy D, where [12] N·
D = I <~l~ilvl~l~l) i=l
This corresponds to a minimization of interorbital electron
interactions. Orbitals ~i so chosen can be expected to show nearly minimum interorbital correlation effects. Such orbitals turn out in general to be relatively strongly
localized in certain parts of the molecule and are therefore
called localized orbitals. Their interest lies in the possible
transferability of localized orbitals between different mole
cules and the eventual transferability of correlation corrections.
There are, of course,other transformation or localization pro
cedures than the one mentioned, and in considering localized
orbitals it is therefore important to always enquire about the
localization criterion.
6. Open-shell SCF methods
6.1. The restricted open-shell SCF method
Suppose we are interested in computing the lowest triplet
state of a molecule. This may either be done by pure CI or
it may be approached by optimizing one-electron functions in
a manner analogous to the closed-shell HF-method. We thus
need an open-shell SCF procedure. If we require our many
electron function from the start to correspond to a definite
spin-state, our open shell procedure will, from that point
of view, be called restricted. The restricted open-shell SCF
procedure in its most widely applied form is mainly due to
Roothaan [13].
54
We write our triplet functions as:
_~ ~~ I CP1 ~1 ••.. CPg~g~m~n I
! ~~ /2 {I CP1 ~1 •••• CPg~gCPm~n I - I q>1 ~ .•.• q>g~gq>n~ I }
~~~ Icp1~1 ..• ·cpg~gq>mq>nl
corresponding to a situation as shown in Figure 18.
We want to minimize
We find for the g
energy g g
3E [2 I ~k + I I {2Jke -Kke} 4--- closed shell terms
k=l k=l e=l
g
+ I {2Jkm-Kkm} + k=l
4--- open shell terms
g
I {2Jkn-Kkn}] 4--
k=l
{closed-open shell
coupling terms
The index g designates the highest doubly filled orbital, the
indices m and n the two singly filled orbitals, the running
indices k and e run over the doubly filled (closed shell)
spatial orbitals only, the indices i and j run over all
spatial orbitals.
The procedure to find a pseudo-one-electron Fock operator
for such an open-shell situation is analogous to the closed
shell case, with the added complication that it proves diffi
cult to get rid of the nondiagonal Lagrangian multiplyers
which couple closed and open-shell orbitals. By some clever
but somewhat tedious algebraic manipulations r13] this may
be achieved, leading, for our particular triplet case, to
55
the open-shell Fock operator of the form: (OS a open shell)
FOS h + 2JT - KT + 2M.r -g
I Jk + 1 (Jm+Jn) 2' k=l
g
I Kk + 1 (Km+Kn) 2 k=l
1 (Km+Kn) 2
g
I Mk + ~ (Mm+Mn) , k=l
2KO where
and where
We tnen have the pseudo-eigenvalue equations
The iterative method of solution (for instance within the
frame of an LCAO expansion) is similar to the closed-shell
case. The corresponding eigensolutions then of course satisfy
the relations
. , The g lowest ~i correspond to optimized doubly filled orbitals
and ~m (where m=g+l) and ~n (where n=g+2) to the optimized singly occupied ones. The energy <3~IXI3~> computed from
these orbitals has attained a relative minimum.
56
In the closed-shell SCF case, by Brillouin's theorem, we have
vanishing matrix elements between the ground state (i.e.
ground configuration) and singly excited configurations
defined within the same set of SCF MO's.
Such a general Brillouin theorem does not hold for open
shells. In our triplet SCF case one can prove that matrix
elements vanish between the minimized triplet 3~n function and m other triplet functions with the same number of singly occupied orbitals, one of which must be either m or n: 3~~ • 3~~ , 3~~~, 3~~~. Figure 18 illustrates this in the case of the six-electron
problem. See also ref. [18].
8
5 5 f if 5 t 4
f I f " f H2f " I "
1 3 3 f H if 3 1 H H 1
I H f 2 2 I 2
1! 7 H 11 H H 3cpt 3cp; 3cp: 34>"5
33 34>""
32
Figure 18
Exercise
l. Consider the triplet SCF state 3~" 3' III 22 3 41. Prove
that (3~jlxI3~~> = <4IFOSI5) 0, where 3~f is defined
in terms of the SCF orbitals of 3~j.
57
In the case of a doublet we have
only one singly occupied orbital,
and the SCF equations are accord
ingly simpler than in the case of
the triplet.
(In reference to Roothaan's paper
[13J: To obtain FOS for the doublet
case requires the same parameters 1 as for the triplet. namely f = 2
a = 1. b = 2; a = 0. ~ = -2.)
t 'Pm
H tpg
H 'Pg-, ,: . H '12 H 'P,
Figure 12 The difference between the Koopmans' energy of the ion (see Section IV.4) and the restricted open-shell SCF energy is
called the reorganization energy.
6.2. The unrestricted open-she'll SCF method
In this method the energy of an arbitrary system of M electrons
with a spin and N electrons with ~ spin is minimized [14J.
The many-electron function ~ is written
~ 1~1~2""~~1~M+2""~M+NI. where the normalization
factor of the Slater determinant of course is {(M+N)!}-1~. The total energy may be ~
E (~I:lCI~>
where
a+~ a+~ a+~ ""
I hii + ~ I I J i j - ~ ( I i i j i
a+~
I i a
I i
means summation over all spin-orbitals with both a; and ~ spin
means summation over all spin-orbitals with only a; spin, etc.
In the double summation the indices take on values independently of each other.
58
Minimization of <~IXI.> leads to separate Fock equations for
orbitals with a and with ~ spin
Fa a IFi
~~
a a £iIFi
£~~
This is accordingly also called the
method of "different orbitals for
different spins".
Although this procedure may take electron correlation to a
higher degree into account than the restricted one, ~ is not
an eigenfunction of S2 and therefore in itself not a physically
~cceptable solution. Once ~ has been optimized, eigenfunctions
of S2 must be projected out of it.
59
V. All-valence MO procedures
As discussed in the introductory chapter, we subdivide the
molecular electrons into groups. From an energy criterion,
we in general merely distinguish between atomic core elec
trons (for instance, ls electrons of second row atoms) and
valence electrons (2s, 2p electrons of second row atoms,
ls electrons of hydrogen). A further subdivision, into a
and n electrons for instance, is of course only possible
in the presence of an appropriate element of symmetry. The
atomic nuclei and the core electrons are assumed to be
frozen into a static core. The B.O.Hamiltonian of the
valence electrons reads:
:!<Val TVal + VVal-Core + VVal-Val
(Val)
+ I I ~>v
1. The Extended HUckel (EH) Method
This is the all-valence electron analogue of the ordinary
HUckel method. Accordingly, one defines an effective one
electron Hamiltonian
:KVal eff
(Val)
I heff{~) The first comprehensive application of this procedure to
organic molecules is due to Hoffmann [15], but similar schemes
were followed previously by Wolfsberg and Helmholz [16] and
others to inorganic molecules.
00
For simple hydrocarbons diagonal atomic matrix elements
hqq 5 <XqlheffIXq) are generally set equal to valence state ionization potentials
Hydrogen
Carbon
(lslhlls)
(2slhI2s)
(2plhI2p)
-13.6 eV
-21.4 eV
-11.4 eV
while for nondiagonal elements hqr a variety of modifications
occur in the literature [15]:
hqr k . Sqr(hqq+hrr ) 1 I . 2
hqr -k • ~r(h~·~r)0 II
h ·h hqr k • S . 2 99 rr III qr (hqq+hrr )
hqr (k-ISqrl) • Sqr(hqq+hrr ) 1 IV . 2
In case IV the off-diagonal elements do not automatically
insure invariance of the eigenvalues with respect to a
rotation of the coordinate axis of reference. k is an ad
justable parameter and is taken to be 1.75 in case I and
between 1.7 and 2.5 in the other cases. As in ordinary HUckel theory, the problem boils down to solving the eigenvalue equation
The overlap integrals Sqr are calculated exactly from Slater
orbitals, with an exponent ~ = 1.625 for carbon and ~H between 1.00 and 1.20 for hydrogen. In ~ electron theory we
have only to consider ~-type overlap (see Figure 20), where
as in the present case we have all possible combinations of
~ and a-type overlap. The neglect of overlap leads here to
meaningless results. The total electronic energy is just the
sum of one-electron energies: E = t bi £i, bi being the
61
occupation number. For molecules with heteroatoms certain
iterative variants of the EH method have been tested to
improve charge distributions. The diagonal matrix elements
are modified by the atomic charges on the respective atoms
until a (limited) self-consistency is achieved.
Figure 20 ~ and a-type overlap
The EH method has proven very successful in predi-cting most
stable molecular conformations and in studying the local and
overall symmetry of molecular orbitals, in particular in
showing the extent of delocalization of a orbitals. The
success in predicting relative conformational energies is
,difficult to explain in a precise way.
To change the conformation of the nuclei implies (within the
B.O. approximation) not only changing the total electronic
energy, but also the purely classical inter-nuclear (or inter
core) respulsion. That iS,we have to include the term
I I zAzBe2 (see page 1) in our energy change, where zA
RAB A> B and zB are atomic core charge numbers. We
thus consider the overall Hamiltonian:
62
(Val)
Xovera11 = ~a1 + VCore- = I h(~) Core ~
It appears a posteriori that with a change of conformation
the effective one-electron Hamiltonian, or rather the total
energy computed from it, varies more like Xovera11 than as XVa1 . However, for very small molecules the conformational
predictive power of the EH method becomes questionable. For
instance, according to the EH method the molecule H2 should
have an energy minimum for united atoms.
The EH-method is n2i well suited for spectroscopic predictions:
Electronic transitions have either to be interpreted in a
pure one-electron scheme, or a CI procedure must be added. The
latter possibility does not prove practical here, however.
2. Electronic population analysis [17J
The electronic population analysis gives a systematic procedure
for the interpretation of LCAO-MO data. Consider a simple MO,
such as
; { Xq being on atom A
Xr being on atom B
and suppose the MO is occupied ny N (= b of sections II.3
and V.1; we here use the notation of Mulliken) electrons.
We then have (all functions are assumed real):
N~2 NC~(XqA)2 + 2NCqCrXqA XrB + NC~(XrB)2
Integrating each term over all space gives
N Nc 2 q + 2NCqCr Sqr + Nc 2
r
Netstomb overlap Net atomic p:;u1atlon population popu1aticn on atom A on atom B
63
We also define the gross atomic population on atom
A as N(Cq2 + c c S ) q r qr and on atom
We now generalize for the case of an MO ~i of a polyatomic
molecule, to which every atom contributes more than one AO.
(~) implies summation over all orbitals on a given atom, (~)
implies summation over all atoms. Thus
c 2 i,qA
and consequently
There are no terms between different orbitals on the same
atom because the corresponding overlap integrals vanish.
Based on the above expression we define:
Net atomic population on atom A for orbital i:
Gross atomic population on atom A for 'orbital i:
c 2 i,qA
The sum of this latter expression over all atoms is just Ni •
Summation over the filled levels i of the net and gross
atomic populations then gives the total net atomic and ~
gross atomic populations on atom A respectively.
3. Semiempirical all-valence calculations. including electron interaction
In the all-valence approach a molecule like ethylene,-which
in section III.l we considered as a two-electron problem-,
has twelve electrons. Considering an AO basis consisting of
the 2s,2Px,2Py,2pz orbitals of the two carbon atoms and the
Is orbitals of the four hydrogen atoms, our SCF matrix will
have the dimension 12 x 12. However, we have seen in the ex
pression for Frp (see page 45) that many-electron integrals
have to be evaluated over all combinations of basis orbitals
which,-in spite of the fact that many integrals will be equal-,
implies the order of 124 integrals. By invoking the ZDO
approximation as defined on page 47, this number is reduced
by a factor of 122. This exemplifies the fact that to keep
calculations on relatively large molecules tractable, approxi
mations of the ZDO-type may be important.
The following sections are devoted to short descriptions of
such simplified computational procedures.
~l. The CNDO (complete neglect of differential overlap) method
The approximations are f18]:
1) The Xp are treated as if they form an orthonormal set;
thus Spq is set equal to bpq •
2) All two-electron integrals which depend on the overlapping
of charge densities of different basis orbitals are
neglected. This means that
(pqlrs)
3) The electron interaction integrals ~pq are assumed to
depend only on the atoms to which the orbitals Xp and Xq
belong and not to the actual type of orbital. Thus ~pq is
set equal to ~AB' measuring an average repulsion between
65
an electron in a valence atomic orbital on A and another
in a valence orbital on B. (The justification for this
approximation will be given below.)
4) The core matrix element hpp contains the interaction energy of an electron in valence orbital Xp on A with the core of A and with the cores of all other atoms B.
It may be written
h 2 \' hpp = (pl- 2m v 2 - VAlp) - L (pIVB\P) B("A)
and is simplified to
as (pIVB\p) is considered to be the same for all valence atomic orbitals on A.
Upp is essentially an atomic quantity, measuring the energy
of an electron in the atomic orbital Xp on the core of A. (See
p.33: Note the difference with the PPP method in defining the core.)
5) Core matrix elements hpq ' where Xp and Xq are different but both belong to A, may in analogy to 4) be written:
hpq = Upq - I (pIVBlq) B("A)
However, due to the mutual orthogonality of S,Px,Py,Pz' Upq is exactly equal to zer~and the remaining terms are
small, so that one sets
hpq = 0 for p " q , Xp and Xq on A.
6) Core matrix elements hpr ' where Xp is on A and Xr is on
B will for simplicitl be considered proportional to the
overlap integral Spr:
hpr = ~o(A,B,RAB) Spr
~o(A,B,RAB) is a parameter dependent on the nature of
atoms A and B, and eventually on their separation, but
66
not on the form of orbitals Xp and Xr.
With these approximations the Fock matrix elements take on the following form:
(A) (B)
Fpp = Upp - L VAB + 2 L DsslAA + 2 L D SIS Il AB - D pp l AA Sl B(+A) s
or, writing 2 DAA = PAA 2 DBB = PBB (see also page 45):
The expression for Fpq (p+q) applies even if p and q are on
the same atom. Then Spq = 0, and lAB is replaced by lAA.
Parametrization of quantities:
Pople and Segal [18] calculate for second-row atoms:
lAA exactly as (2sA 2sAI2sA 2sA)
lAB exactly as <2sA 2sBI2SA 2SB)
Upp from valence-state ionization potentials, using the com
puted value of lAA.
where zB is the core charge
number of B
67
~O(A,B) = ~ (~~ + ~~), where ~~ and ~~ are atomic parameters calibrated on small molecules.
- Other approximations for the y:
th (n ) y AB = Y - a exp -11 RAB
yth is the corresponding exact (analytic) quantity. a, 11, and
n are adjustable parameters. For RAB= 0, Y = yth - a. The para
meter a may be interpreted as a local "correlation term".
- Other approximation for VAB :
VAB = ~ • YAB + P
P refers to the interaction of an electron on A with the
neutral atom B. It is thus a penetration term (and is
eventually neglected). Note the similarity to the PPP
approach (see Section 111.3, p. 33).
4. Invariance of the (exact) SCF eigenvalue problem to unitary basis transformations
We consider a set of closed-shell SCF orbitals ~i' defined in a
basis Xp. In deviation from our usual convention we write
~i I Xp Cpi P
inverting the indices of the coefficients c. In matrix form
(~) (X)(C)
(~) is the row vector of the ~i (i = 1 ••• M), (X) the row vector
of the Xp' and (C) the coefficient matrix of the cpi • This enables us to write the Hartree-Fock equations in a convenient matrix form:
(F)(C) (s)(c)(e)
68
(F) is the matrix Frp of the Fock operator in the basis of the
Xp' (S) the overlap matrix Srp' (e) the diagonal eigenvalue matrix. If {ci } represents the column vector of the coefficients
of ~i' then (F){Ci } = (s){ci}e i , where ei is the (scalar) ith eigenvalue.
We now express our basis (X) in terms of a new basis (X'):
(X) (X' )(u)
Thus (~)= (x')(u)(e) (X' )(e')
The matrices F and S are consequently also to be referred to
the new basis. We find (see below):
(F) (S)
where (U)+ is the conjugate transposed (or adjoint) of (U), and
eFt) and (S') are defined in the basis (X').
For simplicity, we assume both (X) and (X') to form an ortho
normal basis. Thus (e) then is unitary, and (S) becomes the
unit matrix. (e') must also be unitary and consequently also (U).
Thus:
and we obtain
(U)-1 (F' )(u)(e) (e)(e)
Multiplying both sides by (U) and remembering that (u)(e) we obtain
(F' )(e') or equivalently
(F'){ ci} The SeF equations are now defined in the basis (X') and must
(e' ),
lead to the ~ eigenvalues as the ones defined in the basis (X).
It now remains to be shown that (F') with respect to (X') has
the same form as (F) with respect to (X) (see p. 45):
69
Frp [<Xrlhlxp> + I I I C~j Cqj • S q j
{2<xrxslxpxq> - <xrxslxqxp>}]
With xp I x~ Uap Xq a
X* r Ix'*u* y yr X* s y
and noting that by definition
\' L u~s C~j = Cij s
N
and I Cij C~j .. Di~ (= ~ Pi~) j=l
we obtain
Frp I I Uyr Uap [<X~lhlx~> +
Thus
y a
Frp = I I Uyr Uap F~a y a
which completes our demonstration.
I X~ U~q ~
Ix'*u* o os 0
or
70
In case all the coefficients are real, the word "unitary" in
this section may be replaced by "orthogonal", the word "adjoint" by the word "transposed".
Invariance of integrals in the CNDO approximation:
If approximation 2) of the CNDO scheme is applied without
further conditions, the integrals do not necessarily transform
to preserve the invariance of the Fock equations with respect
to unitary transformations of the basis.
Consider the integral (PxA sB I PyA sB> which, according to approximation 2), is neglected. Consider the following unitary
basis transformation: Rotate the (local) coordinate system by 450 clockwise around the z axis.
PxA goes into ..l. (p' + p' ) V2 xA yA 1 ( , ,) PyA into i2 - PxA + PyA
and the corresponding integral goes into
1 (p' s' I p' s' > 1 (p' s' I p' s' > "2 + -xA B xA B 2 xA B yA B
1 (p' s' I p' s' > 1 (p' s' I p' s' > "2 + "2 yA B xA B yA B yA B
of which the first and last terms are not neglected and do not
necessarily cancel. The necessary invariance is, however,
restored by adopting an even cruder approximation, i.e. by
setting
(p' s'lp' s'> xA B xA B (p' s' I p , s ' > yA B yA B
Now the transformed integral becomes zero through cancellation.
In this general way we ensure a pseudo-unitary transformation
of the Fock operator, and it may be shown that the SCF eigen
values remain invariant under the transformation considered.
71
VI. Some special topics
1. Optical activity
A medium is called optically active if the index of refraction
(n) for left (e) circularly polarized light is different from
that for right (r) circularly polarized light:
1'1n ne' - nr + 0
The measurement of this difference as a function of wavelength
A, 1'1n(A), is called optical rotatory dispersion (QfiU). Directly
connected to this effect is the fact that in regions of ab
sorption the extinction coefficient (e) for left and right
circularly polarized light will also differ:
This latter phenomenon is called circular dichroism (QQ). Inside an absorption band ORD will be anomalous, that is,
there will be an inversion of sign. The combined effect of
CD and anomalous ORD inside a region of absorption is called
a Cotton effect.
Optical activity is a molecular effect. A molecule is optically
active when it cannot be superimposed onto its mirror image.
Such a molecule may ll2l have a rotation-reflection axis
Sn (S1 = cr, S2 = i). Many molecules occurring in living
organisms are optically active.
Every transition a ~ bi in an optically active molecule makes
a certain contribution to ~e and An. A CD/ORD spectrum where
these contributions are clearly resolved may appear as shown
in Figure 21. The transition a ~ b1 leads to a positive
Cotton effect, the transition a ~ b2 to a negative one. While
CD may effectively only be measured in regions of absorption,
ORD curves have long tails outside of regions of absorption
which are the superposition of the contributions of different
transitions.
72
Figure 21
Quantum mechanically it may be shown [2,19] that the contri
bution which a given transi~ion makes to the CD/ORD spectrum
is proportional to a quantity called the rotatory strengthffiab :
R is the electric dipole operator (see Section 111.5.) and
M the magnetic dipole operator:
L e ~ -e 2mc I-l I-l
For Simplicity we assume here the summation to go only over
all electrons; we neglect vibronic effects due to the nuclei. ~
e and m stand for charge and mass of the electron, el-l = ~ ~
-ih rj..l x vl-l is the angular momentum operator of the I-lth electron.
73
Im { } means that the imaginary part of the quantity in brackets is taken. The rotatory strength is actually a second
rank tensor, but for a system composed of many identical
randomly oriented molecules one may consider the trace of
this tensor. It is a pseudo-scalar, being the scalar product
of a polar (electric dipole transition moment) and of an
axial vector (magnetic dipole transition moment).
The connection between the rotatory strength and the ex
perimentally determined quantity 6€(A) is given by the
proportionality (-)
For CD and ORO the rotatory strength plays a role formally
comparable to the one of the dipole strength for ordinary
absorption and dispersion (see Section III.5.):
Dab = Re {<~aIRI~b><~bIRI.a>} - S €~A) dA Band
~
Because of the different transformation properties of Rand ~
M under Sn it may be proven that the rotatory strength always
vanishes for systems containing such symmetry elements.
We now wish to show that even with extremely crude wave
functions but which correctly reflect local symmetry proper
ties, a semiquantitative discussion of optical activity is possible.
Case 1: The optical activity of the carbonyl n ~ n* transition
in a ketone (aldehyde). The one-electron energy level scheme
of interest is depicted at left in Figure 22. We assume the
corresponding many-electron states to be well represented by
single-configuration functions. We also consider the highest
filled (n and n) and lowest unfilled (n*) MO's to be markedly
1C'*
n
, I ,-300nm I
-14-0 nm
74
Figure
c o
22
localized on the carbonyl chromophore, as depicted at in Figure 22. In a symmetric ketone, of symmetry C2v' instance, one finds for the transition moments
(nlI!I7r*> 0 (n I iril7r* > + 0 lRn7r* 0
(7rI~I7r*> + 0 (7rliitl7r*> 0 lR7r7r* 0
Thus the n~ 7r* transition, occurring experimentally
right for
at A ~ 300 nm, is magnetic dipole allowed and electric dipole
forbidden, while for the 7r - 7r* transition, occurring at much
shorter wavelength, it is the opposite. The rotatory strength
vanishes in both cases.
Now suppose that we perturb the carbonyl group by introducing
a substituent reducing the overall symmetry to C1 (Figure 23a).
The perturbing substituent will have the effect of slightly
mixing some 7r character to the n orbital:
75
Figure 23a
n' n + An
By first-order perturbation theory
<nIVBln) t>En7l
z
8
8
Figure 23b
where VB is the potential of the substituent R. The rotatory
strength for the n ~ 7l* transition becomes
y
For a given phase of the MO's the sign of Rn7l* will depend on
A. Suppose VR is everywhere positive in space, corresponding
to the potential of the incompletely shielded nuclei of the
substituent (a methyl group, for instance): The sign of the
matrix element <nIVRln) will vary with the position of R as
the sign of the product (n·n), or (y.z), at the position of
R. This leads to a quadrant rule, as depicted in Figure 23b.
If we have just ~ substituent in a given position of cyclo
pentanone (and if we assume the ring to be planar, which in
fact it is not), this result is trivial: By moving R around
76
the C-O bond we merely go from one enantiomer to the other. However, if we move R from a to ~ position in the same
quadrant, we predict that the sign of Rn~ will remain unchanged. If we have several substituents, we may assume their
influence to be approximately additive, and the relative sign
of their respective contributions to be governed by the
quadrant rule.
The quadrant rule is a special case of a series of group
theoretically deducible sector rules [20J. These sector rules can only predict relative changes in sign of the rotatory
strength •. To predict absolute signs, either empirical cali
brations, or more elaborate computations are necessary (see
Case 3).
Case 2:
We consider a molecule as composed of two identical monomers
1 and 2. We suppose that these monomers themselves are
optically inactive, but that they are coupled in such a way
that the dimer is optically active. We suppose furthermore
that the splitting of the two degenerate electric dipole
allowed longest-wavelength transitions may be interpreted
by the dipole-dipole approximation [21J (Figure 24).
---'l~ b_ (A)
--+-~- b+ (8)
_.....1.....1-- a
Figure 24
Then it may be shown that to a degree of approximation which
we presently shall discuss, the rotatory strength of the two
77
longest-wavelength transitions in the dimer is given by
.... .... -+ where R12 = R2 - R1 is the difference of the position vectors of the monomers with respect to some molecule-fixed origin, and -+ .... ~1 and U2 are the electric dipole transition moments of the mono-
-+ .... .... -+ mers: U1 =(a1IRlb1), P2=(a2IRl b 2}. C is a positive constant. As an example we choose 2-2'-diamino-6-6'-dimethyl-biphenyl
(Figure 25). In a simplifying manner we consider the molecule for our purposes to be represented by two coupled aniline
chromophores. We neglect the influence of the methyl substi
tuants and of the bond connecting the rings. We know from
First transition a - b + Lower energy x,z-polarized Symmetry B under C2
Righthanded Chirality
+ Figure 25
Second transition a-b_ Higher energy y - polarized Symmetry A under C2
experiment that in aniline the longest-wavelength transition is
polarized perpendicularly to the C-N bond [22J. From Figure 25
we find~by the coupled oscillator model applied to the composite
chromophore of C2-symmetry and righthanded chirality, in the
order of decreasing wavelength (see also Figure 24):
First transition B-polarized; Rab+ positive.
Second transition A-polarized; Rab_ negative.
This appears to agree with experiment [22J.
78
In our simplified approach we have localized the transition
moments at the geometric centers of the benzene rings. This
is a point of arbitraryness which must be dealt with. In the
present case this choice appears to be admissible, but in
general, if the dimensions of the monomers are comparable to
the distance betwe~n them, then it is not at all obvious
where we should localize these transition moments. There is,
in fact, only ~ point for each monomer where the correspond
ing electric transition moment may be localized as a point
dipole, and for which the formula
JRab C R1 2 • (~2" ~1 )
is exact, otherwise additional terms appear which may be im
portant and may well even make opposite contributions in sign.
Case 3: The molecule cannot be subdivided into a symmetric
chromophore and an asymmetric surrounding as in Case 1, nor
can it be considered as consisting of interacting subgroups
as in Case 2. Rather, the molecule must be viewed as an in
herently dissymmetric entity.
An illustrative example is the molecule shown in Figure 26,
which displays a strong optical activity and long-wavelength
Cotton effects of opposite sign at 294 nm and 263 nm [23]. If we apply the procedure of Case 2 indistinctly, localizing
the electric transition moments ~1 and ~2 in the center of
Figure 26
79
the benzene rings 1 and 2, pointing perpendicularly to the
C-N bonds, the computed rotatory strength vanishes, because ~ ~ ~ ~1, ~2 and R12 are coplanar. This implies that the transition moments should be pointed and localized differently. This
must be done cautiously, or it may easily lead to wrong pre
dictions. In other words, a more accurate assessment of the
quantity 1m {<*aIRI'b><'bIMI*a>} is necessary [24].
A possible procedure, actually leading to correct predictions
with regards to order of magnitude and sign of the Cotton
effects, is the following [10,25]:
1) Compute the SCF ground state (Section IV.I.) and the
lowest excited states by single-excitation CI (Section 111.2.),
assuming local cr-~ separation in all three benzene rings and
invoking the PPP approximation (Section 111.3.). The nitrogen
atoms of the substituents (Figure 26) are considered to be of
the pyrrhole-type and to cintribute two (pseudo-)~ electrons
each. We thus treat the molecule as an inherently dissymetric
(pseudo-) 22~ electron system. The ~pq integrals between all
eighteen carb·on atoms and the two nitrogen atoms enter the
computation. Only the resonance integrals ~pq between nearest
neighbors within a conjugated subunit (N1-benzene ring 1;
N2 -benzene ring 2; benzene ring 3) are taken into account.
The core matrix thus has the aspect given in Figure 27.
2) From the semiempirical wavefunctions the rotatory strength
of the longest wavelength transitions are then evaluated.
In practice it proves necessary to compute the electric dipole
transition moment in the dipole velocity form. For exact
eigenfunctions 'a and $b of the same Hamiltonian we have the ~
identity (see p. 39 for definition of R):
where i = L ~~ . For the rotatory strength we then obtain ~
benzene N/ 1
~ \~ where L ;: L e~
~
80
benzene 2 3
Figure 27
Expressing the wavefunctions in terms of configurational
functions
. •
we find (see Section
(wa lwl1j1b> ... Y2I B~ ik
~ Y2I B~ (tbIL1*a> .... ik
III.5.):
(Cfli I~ I Cflk > and
-+ (Cflkl e 1Cfli>
The configurational coefficients B~ are assumed real, likewise
the SCF-MO's Cfl i • As t;: -ill :;,,,~ and as (Cflkl:;'x~ICfli> =
- <Cflil:;'x~ICflk>' we finally obtain
81
The approximations inherent in this expression originate in
the approximate nature of ta and Wb. k Returning to our example, the ~i and B. are computed as de-~ -+
scribed under 1). The matrix elements <~ilvl~) and -+ ..
<~ilrxvl~) are numerically evaluated without further approximations. They reduce to integrals between atomic
orbitals which, in the case of Slater orbitals, boil down to
the evaluation of linear combinations of overlap integrals
[25]. The matrix elements of ~ A~ depend on the origin of the
coordinate system; the matrix elements of ~ do not. It may be
shown that Rab , as computed above, likewise is origin-inde
pendent, which of course it should be. However, as mentioned,
the electric dipole transition moment must be computed in the
dipole velocity form [26J.
Using standard PPP parametrization [22J and taking into
account 99 singly excited configurations we obtain for the
longest-wavelength Cotton effects of our triptycene derivative
(Figure 26):
291 nm
285 nm
264 nm ,
R01
R02
-0.44.10- 36 cgs
+0.33.10- 36 cgs
+0.21.10- 36 cgs
This appears to agree with experiment as to order of magnitude
and sign.
2. Selection rules for electrocyc1ic reactions and cyc10-addition reactions
Molecular orbital theory in its simplest form, in particular
the EH approximation [15], has provided a brilliant means of
rationalizing and interpreting the regularities encountered
in concerted organic reactions. The corresponding rules, now
generally called Woodward-Hoffmann-ru1es [27J. grew out of
the necessity to rationalize empirical evidence such as the
82
following:
Thermal (~) ring closure of butadiene proceeds by a conrotatory
movement of the substituents, photochemical (hv) ring closure
goes in a disrotatory way-. For hexatriene it is the opposite.
Already prior to Woodward's and Hoffmann's systematic investi
gations, Fukui and Oosterhoff had independently [28] suggested
that the course of such reactions might be connected to the symmetry of the highest occupied orbital of the polyene. The
2Pn AO's of the terminal carbon atoms may be thought of as combining with the proper phase to form a bonding a orbital.
In butadiene in the ground state (thermal path) the highest
filled orbital is n2 (Figure 28), while in the first excited
state (photochemical path) it is n3.
From a more general point of view it became clear that such
selection rules could be better interpreted by looking at the
83
Figure 28
totality of participating electrons [29J. Assuming that the
states of the reactant and of the product may be characterized
by a common and relevant symmetry element, correlation dia
grams can be drawn as shown in Figures 29a,b and 30a,b for
electrocyclic reactions, and 31a and 31b for cycloaddition
reactions. The following points are of importance:
1) During the transformation at least one element of (overall
or local) symmetry is maintained. The states of reactant,
product and most plausible transition state may be
characterized by it.
2) The symmetry element of importance must bisect the bond(s)
which is (are) being formed or broken.
3) In correlating the states of reactant and product the
"noncrossing rule" holds: Lines correlating states of
same symmetry may not cross.
84
4) For simplicity it is assumed that the states of interest
can be described by single configurations, defined in a
set of appropriate one-electron MO's. The one-electron
states are of course also caracterized by the same symmetry
element(s) and correlate in a similar way as the overall
many-electron states. The energetic sequence of the one
electron states and their occupation determines the
energetic sequence of the many-electron states.
5) There are cases where the correct correlation diagram cannot
be established unambiguously by inspection. In such cases
a series of EH calculations along the reaction path may
illustrate how the molecular orbitals gradually evolve.
Two remarks must immediately be added:
- The present MO theoretical description and interpretation
does not automatically lead to the most compact formulation
of the selection rules, as the practical chemist seeks them.
On this question there also exists a vast amount of
literature [30J.
- The simplicity of the elementary molecular orbital approach
invites further refinement. Until now it appears that more
sophisticated treatments basically confirm the results ob
tained from the simpler picture [31J.
We now turn to elementary examples:
Figure 29a illustrates the electrocyclic reaction of butadiene
over a conrotatory path. The symmetry element C2 is maintained
throughout and the one-electron states correlate as shown. The
transformation properties of the corresponding many-electron
states, denoted by big letters A (symmetric) or B (anti
symmetric), is immediately deducible from the symmetry,
designated by small letters a or b, of the MO's occupied by
electrons participating in the reaction. In the thermal case
we see that the ground state (ground configuration) of buta
diene correlates with the ground state (ground configuration)
Th
erm
al
<l-
--C
>
1C+a
7r3
b
7r2a
T1
\.
1C, b
T±
n:2
_2
1 .I&
.2
A
/'
"-
Bu
tad
ien
e <
I----t
> C
yclo
bu
ten
e
Co
nro
tato
ry
pa
tn
Sym
met
ry C
2
Pho
toch
emic
al
ba*1
a1C*
1
6.
b1C I
aa
I
a2n:
2
A
--/-
f>
I
1C+ a
1C3 b
----r-
/
1C2a
" 1C
, b
Ti /'
n:/n
: 2 :1l
"3
B
rba*
i I a1
C*1 I I I I I
-" t1
b1
C I I
'" t a
a
.
a n:
2a
* B
Fig
ure
2
9a
11;+ a
1C3b
7r2
a 'V
1C, b
n: 1n:
1n:,
B
<r+
-b
att
.......
T
a1C
*
co
C11
'\.
/ '(
b1C
" T
I a
a
a2n:
n:*
B
86
Energy
U7r2U*
Reaction path
Figure 2gb
of cyclobutene. This implies (Figure 29b) that no particular
potential barrier is to be expected on going from one species
to the other. In the photochemical case. on the other hand.
the situation is different: Starting with the first excited
state (configuration) of one species we end with the second
excited state (configuration) of the other (Figure 29a). The
configuration ~f ~2 ~3 of butadiene correlates with cr ~2 cr*
of cyclobutene, and for the reverse photochemical reaction
cr 2 ~ ~ of cyclobutene correlates with ~1 ~~ ~4 of butadiene.
Both paths involve an increase in energy. Now,because all of
these four states are of the same symmetry B with respect to
C2, the adiabatic paths will interact due to the "noncrossing
rule". This implies that if we start out with butadiene in
the configuration ~~ ~2 ~3 we will not end up with cr ~2 cr*
as deduced from Figure 29a, but with cr 2 ~ ~. However, the
Bu
tad
ien
e <
l--C
> C
yc/o
bufe
ne
Dis
rota
tory
pa
th
Sym
met
ry C
s T
herm
al
~
<J-
f----
a"u*
I a"
u *
7C. a"
I
7C. a
"'-
--.
_ .
I I ~
a"1r
* I
a''1
r *
1r 3
a'
/ I
1r3 a'
t I
~
1r2a
" T
I ---
a'7('
7r1a
' ti
----
n a'
u
1I:{1
I:J
A'
q2
11
:*2
A'
7r 2aJ
l -
" I
I a'
7r
7r1 a'
+:l
; ----
fi a'
u
1I:{1
I:f
A'
q211
:2
A'
Fig
ure
30a
Pho
toch
emic
al
<J--
--t>
a"u*
7('
.a"-
--
_....j.
--a"
7r*
1r3a
'-......
. -
7r2a
" T
I
" a'
1r
7r1 a'
fi
----
ti
a' u
2 11
:, 11
:211:3
A"
q211
:11:
*
A"
!S
88
Energy
Reaction path
Figure 30b
reaction will have to proceed over a high potential barrier,
and this course is accordingly very unlikely. For I(t con
rotatory reaction path the thermal course will consequently
be preferred over the photochemical course.
In a disrotatory reaction path a plane of symmetry is main
tained. One-electron states and many-electron states are
characterized with respect to it by the respective symbols
a' or A' (symmetric), a" or A" (antisymmetric) (Figures 30a,b).
The correlations are established accordingly. Here we see
that it is the thermal course which leads over a high potential
barrier. In the one-electron picture (Figure 30a) the ground
configuration of one species apparently correlates with a
doubly excited configuration of the other. Although the "non
crossing rule" comes into play (Figure 30b), it is to be de
duced that the transition state in the thermal course will
89
nonetheless be of relatively very high energy. On the other
hand, in the photochemical case, the lowest singly excited
configuration of reactant always correlates with the same
singly excited configuration of the product. In the dis
rotatory reaction path the photochemical course will consequently be favored over the thermal course.
We now turn to cycloaddition reactions and consider the
example of the addition of ethylene and butadiene to give
cyclohexene. The course of this reaction proceeds in such a
way that a plane of symmetry is maintained.
~I * I 1{ I
I
q*.(i\, 2~
I
utOo
Figure 3la
1{*
90
In accord with group theoretic usage we characterize the one
electron states by a ' and a" with respect to the relevant plane
of symmetry. as in the previous example. Figure 31a shows at
left the three highest occupied orbitals of the reactants
butadiene+ethylene in the proper relative energetic sequence
1t1 (a ' ), 1t(a l ), 1t2(a") and the three lowest unoccupied orbitals
in the corresponding relative sequence 1t3 (a '), 1t*(a" ). 1t4 (a").
In Figure 31a at left the three highest occupied and three
lowest unoccupied orbitals of the product cyclohexene are drawn
--a" ut
--- a" a' --- U,*
7C*---a" ~ II ~ a--7C3 -- a'
Figure 31b
91
in a simplified way, also in the proper energetic sequence
a1 (a'), 02(a"), n(a'); n*(a"), aHa'), a~(a"). The correspond
ing correlation diagram for the one-electron states is shown
in Figure 31b. Considering the occupation of the orbitals,
we see that the ground configuration of the reactant corre
lates with the ground configuration of the product. making
the reaction thermally allowed.
The photochemical course, starting from the lowest excited
configuration ~f ~2 ~2 ~3 of the reactants, correlates with
the higher excited configuration 0* o~ ~ af of the product
and is therefore energetically unfavorable. The question is of
interest, if the reaction could not proceed from the photo
excited ethylene, instead of the photoexcited butadiene. The
configuration ~~ ~ ~~ ~ indeed correlates with the lowest
excited configuration of the product, of a~ ~ ~. However,
it is possible that as soon as the reactant molecules inter
act,the state ~f ~ ~~ ~ internally converts by vibronic
coupling to the lower excited state ~~ ~2 ~2 ~3, thereby
impeding the further course of the reaction.
For a discussion of other concerted organic reactions, such
as sigmatropic or cheletropic reactions, the reader is re
ferred to the literature [27,30J.
Exercise: Draw the state correlation diagram for the cyclo
addition reaction ethylene+butadiene, based on Figure 31b.
3. Molecular orbital theory with periodic (cyclic) boundary
conditions
This form of molecular orbital theory provides a means of
studying in the tight-binding approximation the electronic
structure of polymers built from sequentially repeating
subunits [32J. It is assumed that, neglecting end effects,
cyclic boundary conditions may be applied. We consider a
92
polymer consisting of N· monomers and assume that per monomer
v valence orbitals (atomic orbitals) have to be taken into
account. The number of electrons per monomer is ve. Further
more, we denote the monomers by the indices p, p', q, q',
the AO's within the monomers by s, s', t, t'. We characterize
the polymer MO's by double indices jm, j'm', en, e'n', where
j, j', e, e' designate the symmetry of the MO under the
cyclic point group CN•
The s th atomic orbital within the p th monomer is written
Xps. Let us choose the origin of the molecule-fixed coordinate
system to coincide with the center of a particular AO, Xs = Xs (t). The corresponding AO Xs (t - Er) will be centered in the
point Er, Xs (t - 2Er) will be centered in the point 2Er, etc. If
Er is the primitive translation vector of a linear polymer we
may write
Xs(t) Xos
Xs(t-Er) X1 s
Xs(t-2~) X2S
Xs (t - pit) Xps
Considering the periodic boundary conditions or formal cyclic
symmetry we in general write a molecular orbital, extending
over the whole polymer, as [33J: N v
L j wjpc. X .-' Jm, s ps
p=l s=l
or in bracket notation
I jm) Ips)
p s
where w = exp(2~i/N). In a given polymer the Xps will have to
be appropriately chosen, so as to conform to the particular
(for instance helical) geometry. From the above relations one
93
finds
N \I
I I wjpc X jm,s (p-l)s
p=l s=l
p=l s=l
which, due to the cyclic properties
and consequently
The function must indeed have the same value after N elementary
translations.
The HUckel/Extended HUckel approximation:
See Chapter II and Section V.l.
We define symmetry orbitals (in bracket notation):
N
I js) L wjp Ips)
I j,j' = 1 •••• N
p=l or equivalently, for N even:
N j,j' = O,±l, ... ± (N/2-1), N/2
\j's')= I j 'p' \p's') for N odd:
w j,j' = O,±l,
p'=l ••• ± (N-l)/2
and express the eigenvalue equation in the basis of these
orbitals
\(jslhlj's') - €(jslj's')\ 0
94
This is equal to
II {(PSlhlp's') - e(pslp's')} o
The indices s, s' go from 1 to v. It may be shown [34] that the elements of this secular determinant vanish, unless
j = j'. The eigenvalue problem, originally of order N'v,
thus factorizes into separate equations of order v for
every value of j.
Example: A cyclic chain of N "ethylene" molecules (see
Figure 32). We have s, s' = 1,2
p-1
{J' {1
1 2
(J' 1
p
(J
2
Figure 32
p+1
(J' (J {J' 2
Invoking the ZDO and nearest-neighbor approximation, as in
ordinary HUckel theory, we find
(pllhlpl)
(pllhlp2)
(pllhl (p-l)2)
(p2Ihlp2) a
13 (p2Ihl(P+l)1) = 13'
Adding the contributions for p 1,2 ••• N and then dividing
every element by N we obtain for the secular equation
95
o a - e
Leading to the solutions
{ 27fj}1/2 e a ± ~2 + ~'2 + 2~~' cos N
For ~' = 0, the formula reduces to the HUckel energy levels
of independent ethylene molecules. For ~ = ~' we get
a ± 2~ cos 1t With N=3 we correctly obtain the HUckel energy levels of
benzene (see page 10), j taking on the values 0, ±l.
The SCF formulation:
See Sections IV.l. and IV. 3 ( Attention: The symbol N there and) here has a different meaning.
The Hartree-Fock equation in the basis of symmetry orbitals
may be expressed as
I (jsIFlj's') - e(jslj's') I 0
(jsIFlj's')
N 'Ve/2
(jslhlj's') + L I {2 (js enlj's' en) - (js enlen j'S')} e=l n=l
The summation goes over all occupied polymer orbitals. Here
we assume that there is no overlapping of the energies of
bands for which n $ 'Ve/2 and of bands for Which n > 'Ve/2.
We admit that we have the situation shown in Figure 33a and
not the one illustrated by Figure 33b. A polymer MO len) is written
len) I I weqcen,t Iqt) q t
96
The symmetry index e - like j - takes on the values
e = 1, 2 ••. N or, equivalently, e = 0, ±l, ••• ±(N/2-1), N/2 •
It may be shown [34J that matrix elements of the Fock operator
between symmetry orbitals vanish, unless j' = j. With s,s' =
1, 2 .•. v, the Fock matrix consequently factorizes into N
submatrices of order v. Expanding the Fock operator we find:
(jsIFljs') = [P~, P,~, wj(p'-p) (pslhlp's') +
N N N N
c* c , en,t en,t I I I I wj(p'-p)+e(q'-q) •
e=l n=l t=l t'=l p=l q=l p'=l q'=l
• {2 (ps qtlp's' q't'> - (ps qtlq't' pIS'>}]
This formula is exact within the Hartree-Fock approximation.
It is indeed rather cumbersome to evaluate this expression
for large N, in particular the two-electron part. The fall-off
monomer polymer
1 2 Ve }N
~ Ve -1 ti :::::::::=:l~ } N
Figure 33a
monomer polymer
N
N
Figure 33b
97
of the corresponding integrals with distance may be taken
into account to simplify it, as indicated in [34J.
Within the frame of a semiempirical approach the ZDO approxi
mation is useful (see Section IV.2). The electron repulsion
integrals (ps qt\p's' q't') are then neglected, except when
ps = p's' and qt = q't'. SimilarlY only integrals of the form
(ps qtlq't' p's') are retained for which ps = q't' and
qt = p's'. We remember that sand s' are fixed indices for
a given matrix element, while t and t' are running indices.
The two-electron part then reduces
for the diagonal element (jsIF\js) to:
N Ve/2
I I {2 I c1n,t Cen,t I L (ps qtlps qt) e=l n=l t p q
- c* c \ \ w(j-e)(q-p) en,s en,s L L (ps qslps qs)}
p q
and for the nondiagonal element (js!F\js') to:
c* en,s c , ';\ w(j-e)(q-p) (psqs'\psqs') en,s '-" L
p q
The computation of optical properties [34J: The CI matrix also
factorizes according to the irreducible representations of the
group CN (for N even):
A, E+1 , E_1 , ••••• E+ j , E_ j ••••. E+(N/2_l)' E_(N/2_l)' B
One starts from a total of N'v polymer MO's, of which
N • ve/2 are filled. Consequently N·.,f possible singly excited
configurations may be constructed, where
j( = N' v2e ('J _ ~e)
98
With respect to their symmetry, these singly excited con
figurations are evenly distributed over all the irreducible representations, as long as we assume a situation as shown
in Figure 33a. Consequently there should be J(' singly excited
configurations belonging to each of the N representations
A(j=O), B (j=N/2) and sub-representations E+j , E_j' As the
dimension of the polymer grows, the computational labor of
diagonalizing these N large matrices will rapidly become
immense. To limit the expense in calculating optical spectra,
one should then give priority to the excited states to which
transitions from the ground state are electric dipole allowed.
The corresponding selection rules are therefore of immediate
interest. We assume that higher retardation effects may be
neglected. To derive the selection rules it is essential to
distinguish between the formal cyclic symmetry CN - which we
assume always to apply in the sense described - and the actual
geometry.
One finds:
a) For a linear geometry transitions are allowed when j' j.
This implies that there is only one CI submatrix of
dimension Jrbetween singly excited configurations belong
ing to the irreducible representation A to be considered.
b) For a cyclic geometry transitions parallel to the symmetry
axis of r~tation are allowed when j' = j. In-plane transi
tions are allowed when j' = j±l. This implies considering
the CI matrices beionging to the irreducible representations
A, E+I and E_I •
c) For a helical geometry the selection rule j' = j holds in
the case of transitions parallel to the helical axis. For
components polarized perpendicularly one obtains j' = j±M,
where M = N/~ and ~ is the number of monomers per helical
turn. In general, ~ will not be an integer. However, M
99
must be an integer. Therefore a clear-cut selection rule
will only hold for values of N which correspond to one or
several translational identity periods along the axis of
the helix. The CI matrices to be considered will belong to
theirreduoible representations A and E+M, E_M•
Some References
[lJ M. Born and R. Oppenheimer, Ann. d. Physik 84. 457 (1927).
[2J H. Eyring, J. Walter and G.E. Kimball, "Quantum Chemistry",
John Wiley, New York 1963.
[3] a) E. HUckel, Z. Physik IQ, 204 (1931).
b) R. Daudel, R. Lefebvre and C. Moser, "Quantum
Chemistry, Methods and Applications", Interscience,
New York 1959.
c) A. Streitwieser, "Molecular Orbital Theory for Organic
Chemists", John Wiley, New York 1962.
d) L. Salem, "The Molecular Orbital Theory of Conjugated
Systems", Benjamin, Inc., 1966.
e) E. Heilbronner and P.A. Straub, Table of HUckel
Molecular Orbitals, Springer-Verlag, 1966.
f) E. Heilbronner and H. Bock, "Das HMO-Modell und seine
Anwendung, Grundlagen und Handhabung", Verlag Chemie,
Weinheim 1968.
[4J A. Carrington and A.D. McLachlan, "Introduction to
Magnetic Resonance", Harper and Row, Hew York 1969, p. 89.
[5J R.G. Parr, "Quantum Theory of Molecular Electronic
Structure", Benjamin, Inc., New York 1963.
[6J a) E.U. Condon and G.H. Shortley, "The Theory of Atomic
Spectra", Cambridge University Press, Cambridge 1963, p. 169-174.
b) J.C. Slater, "Quantum Theory of Atomic Structure",
Vol. I, Mc Graw-Hill, 1960, p. 291-295.
[7J J.A. Pople, Proc. Phys. Soc. A 68, 81 (1955).
[8J R. Pariser and R.G. Parr, J. Chern. Phys. £1, 466, 767 (1953) .
101
[9J W. Moffitt, J. Chern. Phys. 22, 320 (1954).
[10J J. Fiutak, Canad. J. Phys. 41, 12 (1963); see also
R.E. Geiger and G. WagniE'lre, in "Wave Mechanics, the first fifty years", ed. W.C. Price, S.S. Chissick,
T. Ravensdale, Butterworths, London 1973, Chap. 18.
rllJ a) R.S. Mulliken, J. chim. phys. 46, 497, 675 (1949).
b) C.C.J. Roothaan, Rev. Mod. Phys. ~, 69 (1951).
112J C. Edmiston and K. Ruedenberg, J. Chern. Phys. ~, S97
(1965) .
[13J C.C.J. Roothaan, Rev. Mod. Phys. ~, 179 (1960).
[14J J.A. Pople and R.K. Nesbet, J. Chern. Phys. 22, 571 (1954).
[15J R. Hoffmann, J. Chern. Phys. Z2, 1397 (1963); see also W. Hug and G. Wagniere, Tetrahedron~, 631 (1969).
[16J M. Wo1fsberg and L. Helrnho1z, J. Chern. Phys. 20, 837
(1952).
[17J R.S. Mulliken, J. Chern. Phys. ~, 1833,. 1841 (1955).
[18J J.A. Pople, D.P. Santry and G.A. Segal, J. Chern. Phys.
~, S129 (1965); J.A. Pople and G.A. Segal, ibid. ~,
S136 (1965); see also M. Jungen, H. Labhart and
G. Wagniere, Theoret. Chirn. Acta i, 305 (1966);
J.M. Sichel and M.A. Whitehead, Theoret. Ohirn. Acta 1, 32 (1967); R.J. Wratten, Chern. Phys. Letters 1, 667
(1968).
[19J L. Rosenfeld, Z. Phys. 52, 161 (1929); E.U. Condon,
Rev. Mod. Phys. 2, 432 (1937); E.U. Condon, W. Altar
and H. Eyring, J. Chern. Phys. 2, 753 (1937).
[20J W. Moffitt, R.B. Woodward, W. Klyne and C. Djerassi,
J. Am. Chern. Soc. ~, 4013 (1961); J.A. Schellrnan,
J. Chern. Phys. 44, 55 (1966); A. Moscowitz, Adv. Chern.
Phys. 1, 67 (1962); J.A. Schel1man and P. Oriel,
102
J. Chern. Phys. 21, 2114 (1962); G. Wagniere, J. Am.
Chern. Soc. 88, 3937 (1966).
[21J I. Tinoco, Adv. Chern. Phys. ~, 113 (1962); J.A. Sche11-man, Accts. Chern. Res. 1, 144 (1968).
[22J R.E. Geiger and G.H. Wagniere, in "Wave Mechanics, the
first fifty years", ed. W.C. Price, S.S. Chissick,
T. Ravensdale, Butterworths, London 1973; H. Labhart
and G. Wagniere, He1v. Chim. Acta 46, 1314 (1963).
[23J J. Tanaka, F. Ogura, M •. Kuritani and M. Nakagawa,
Chimia 26, 471 (1972).
[24J A.M.F. Hezemans and M.P. Groenewege, Tetrahedron~,
1223 (1973).
[25J W. Hug and G. Wagniere, Theoret. Chim. Acta 18, 57
(1970); G. Wagniere, in "Aromaticity, Pseudo-Aromaticity, Anti-Aromaticity", the Jerusalem Symposia on Quantum
Chemistry and Biochemistry, III. The Israel Academy of
Sciences and Humanities, Jerusalem 1971, p. 127;
G. Blauer and G. Wagniere, J. Am. Chern. Soc. 21, 1949
(1975) •
[26J W. Moffitt, J. Chern. Phys. ~, 467 (1956).
[27J R.B. Woodward and R. Hoffmann, J. Am. Chern. Soc. §I, 395 (1965); R. Hoffmann and R.B. Woodward, J. Am. Chern.
Soc. §I, 2046 (1965); R.B. Woodward and R. Hoffmann,
"Die Erha1tung der Orbita1symmetrie", Verlag Chemie,
Weinheim 1970.
[28J K. Fukui, T. Yonezawa and H. Shingu, J. Chern. Phys. 20,
722 (1952); K. FukUi, T. Yonezawa, C. Nagata and
H. Shingu, J. Chern. Phys. 22, 1433 (1954); L.J. Oosterhoff, cited in E. Havinga and J.L.M.A. Sch1atmann,
Tetrahedron 16, 151 (1961); see also ref. [27J.
103
[29J H.C. Longuet-Higgins and E.W. Abrahamson, J. Am. Chem.
Soc. §i, 2045 (1965).
[30J N.T. Anh, "Les Regles de Woodward-Hoffmann", Ediscience,
Paris 1970.
[31J J.J. Mulder and L.J. Oosterhoff, Chem. Commun. 121Q, 305; E.B. Wilson and P.S.C. Wang, Chem. Phys. Letters
12, 400 (1972).
[32J J. Koutecky and R. Zahradnik. Collection Czech. Chem.
Commun. ~. 811 (1960); T.A. Hoffmann and J. Ladik,
Advan. Chem. Phys. 1, 84 (1964); J. Ladik and K. Appel,
J. Chem. Phys. 40, 2470 (1964); A. Imamura. J. Chem.
Phys. ~, 3168 (1970); K. Morokuma, Chem. Phys. Letters 2. 129 (1971); J.-M. Andre, G.S. Kapsomenos and G. Leroy,
Chem. Phys. Letters Q, 195 (1971); J. Bacon and D.P.
Santry, J. Chem. Phys. 2§, 2011 (1972).
[33J F. Bloch, Z. Physik ~. 555 (1928).
[34J G. Wagniere and R. Geiger, Helv. Chim. Acta 56, 2706 (1973) •
SUBJECT INDEX
Ab initio calculations 3.45
All-valence calculation. semi-
empirical 59.64 Alternant hydrocarbon 12
Analysis. population 62
Aniline chromophore 77
Approximation. nearest-
neighbor 7.32
-. ZDO 7.30,40,47.64.97 Atom-atom polarizability 14
Atomic density, HUckel 11
-. population 62
Basis function 5 Basis transformation 67.68
Benzene, electronic spectrum
37 -, HUckel model of 9
• PPP model of 34-37 -, transitions 37.41 Bond-bond polarizability 14
Bond order 12
Born-Oppenheimer approximation
1.2
Brillouin theorem 49,56
C2 symmetry 77
Calibration of resonance integrals 8
Canonical SCF orbitals 52
Carbonyl group 74
Charged sphere approximation
31 Chirality 77 Circular dichroism CD 71
Closed shell problem 42
CNDO. Fock matrix 66 method 64
Commutator with spin 18
Complex benzene MO's 10,35
Concerted reaction 81
Configuration. electron 12.16, 84
- interaction. CI 26-29.35.80.
97 - mixing, CI 26-29.35.80.97 Conformation. prediction of 61
Conrotatory electrocyclic re-
action 82/83
Core electrons 59
- integral. PPP 24,32
- matrix elements. CNDO 64
- operator 25 - resonance integral 25 Correlation diagram 84
- energy 50-52
Coulomb integral a. HUckel 7 -, two-electron 29.42
- operator J 48.54 Cotton effect 71
Cyclic symmetry 37.91
Density, atomic 11 - matrix, first order 12,45 Determinant, secular 6,17,45,
47,93,94 -, Slater 15,26-29 Different orbitals for differ-
ent spins 58 Dipole-dipole approximation 76 Dipole strength 39 - vector form of electric
transition moment 79 - velocity form of electric
transition moment 78 Direct product of representa
tions 40 Disrotatory electrocyclic re
action 82,83 Doublet state 57
Effective Hamiltonian 59 Electric dipole operator 38,
39,78,79 - dipole transition moment
38,39,78,79 - dipole transitions,
selection rules 40, 97 Electrocyclic reaction 81 Electron affinity, atomic 24 - configuration 12,16,84 - repulsion integral, CNDO 64 - repulsion integral, PPP
30-32 Electronic population analy
sis, Mulliken 62
105
Electrostatic orbital self-
energy 53 Energy levels of benzene 37 - of ionization 50 - of reorganization 51 Equation, secular 6,17,45,47,
93,94 Ethylene, HUckel 7 -, PPP 15 Exchange integral 29,42 - operator K 48,54 Extended HUckel approximation
59,93
Fock matrix elements, general
closed shell 45,59 - matrix elements, general
restricted open shell 55,56 - matrix elements, CNDO 66 - matrix elements, with cyclic
symmetry 96 - matrix elements, ZDO 47,97 - operator F, general closed
shell 45 - operator, general restricted
open shell 55,56 - operator, unrestricted open
shell 57,58 - operator, invariance 48
Gross atomic population,
Mulliken 63
Hamiltonian, nonrelatiVistic,
Born-Oppenheimer 1
, semiclassical radiation 38
Hartree-Fock limit 45
- method, see also under Fock
42,95
Hermitian matrix 48
Heteroatoms, HUckel parameters
for 13
HUckel approximation 4,59,93
- method 4 - method, extended 59
Inherent dissymmetry 78
Integrals,
semiempirical evaluation of
- CNDO 64-67
106
- extended HUckel 60
- HUckel 7-9,13,14
- PPP 30-34 Invariance, to basis trans-
formation 67-70
Ionization energy 50
- potential, . valence state 24
Irreducible representation of
point groups 22,40,97
J, see Coulomb operator 48-54
K, see Exchange operator 48,54
Koopmans 1 "theorem" 50
Lagrangian multipliers 44,48,
54
LCAO-MO, Ritz variational
method 5
Limit, Hartree-Fock 45
Linear combination of atomic
orbitals, see LCAO-MO 5
- combination of configura
tions, see Configuration
interaction 26-29
Localization of transition
moments 78
Localized orbitals 52
Magnetic transition moment 72
Many-electron theory 15,42,59
Matrix-elements, CI 29,36,49,
56,97
- elements, Fock 47,66,97
Molecular orbitals
-, analysis of 62,63
CNDO 64-67
-, Cyclic 10,35,91-97
-, Extended HUckel 59-62
-, HUckel 7-14
, MO see LCAO-MO 5
, PPP 30-38
, SCF 42-46
-, of benzene 10,35
of ethylene 8,16
-, in concerted reactions 81-91 Moment, electric transition
39,72
Nearest-neighbor approximation
7,32
Net atomic population 62,63
Noncrossing rule 86
n~*-transition 73
Off-diagonal Lagrangian
multipliers 44,48,54
One-electron integrals
- approximation of 24-26
- CNDO 64-67
- Extended HUckel 60
- HUckel 7-9, 13,14
- PPP 30-34 Open-shell SCF method,
restricted 53
unrestricted 57 Optical activity 71
Orbital
-, symmetry- 11,93 -, symmetry of 22,81-91
Orbitals
-, see LCAO
-, see Molecular orbitals
-, canonical SCF 46,52
-, localized 52
ORD, see Optical activity 71
Orthogonal transformation of
basis 67-70 Overlap population 62
Pairing of electronic states
13
107
Parametrization, see Integrals
Pauli-principle 15
Periodic boundary conditions
91-99 Photochemical cycloaddition
reaction 91
- electrocyclic reaction 87,88
Pi-electron approximation 4,30
- separation, overall 4,15,30
- separation, local 79
n~*-transitions
-, in benzene 35-41
-, optical activity of 76-81
Point group symmetry 22,40,84,97
Polarizability atom-atom 14
- bond-bond 14 Population analysis, Mulliken
62
-, atomic 62
-, overlap 62
PPP approximation 30
- core 34 Pseudo eigenvalue equation, SCF
49,55 Pyrrhole 34
Quadrant rule 75
Reactions, concerted 81-91
Real form of benzene orbitals
10
Reorganization energy 51,57 Representation, see
Irreducible representation
Resonance energy 12
Resonance integral ~,
- HUckel 7 - PPP 25,26 Restricted open-shell SCF
method 53 Ritz variational principle 5 Rotational strength, see
Rotatory strength 72 Rotatory strength 72
SCF method, see self consistent field
- equations, open shell 53-58 - equations, closed shell
42-53 Sector rule 76
108
Secular equation,
see Equation
Determinant
Selection rule for electric
dipole transitions 40,97-99 Self consistent field SCF,
see Hartree-Fock method Fock matrix
Molecular orbitals SCF
Semiclassical radiation theory
38,72 Semiempirical procedures,
see Approximations
Integrals
Molecular orbitals
Separation, Born-Oppenheimer
1,2 , a-n 4,15,30,79
a electrons 4,59
Singlet function 19,26 Singly excited configurations
see Configuration interaction
Brillouin theorem
Slater-Condon rules Slater-Condon rules 28 Slater determinant 15,26-29 Spectroscopic calibration of ~
9 Spectrum, computed of benzene
37 , circular dichroism 71-81
-, computed of ethylene 22 Spin eigenfunctions of Sz, S2
18 Spinorbital 15 Strength, dipole 39 Substituent effect 74 Symmetry
-, spatial see Point group
symmetry
- of excited benzene states
37,41 - orbitals 11,93 - rules for concerted re-
actions 81-91
Thermal reaction, concerted
84,88 Thermochemical calibration of
resonance integral 8 Total gross atomic population
63 - net atomic population 63 - SCF energy 49
Transition moment
-, electric dipole 39,72,79
-, magnetic dipole 72
Transition probability,
electric dipole
see Dipole strength 39
Triplet function 19,27,54
Two-electron integrals 19,23,
30
see also Coulomb integral
Exchange integral
Uniformly charged sphere
approximation 31
Unitary transformation of
basis 48,67-70
109
Unrestricted open shell SCF
method 57
Valence, all-, procedure 59
- electrons 3 - state ionization potential
24,33
Variational principle
-, Ritz 5,17
-, SCF 42
Vibrational wavefunction 2
Zero differential overlap
approximation ZDO 7,23,30,
36,40,47,64,97
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