Lecture 5: First principle calculation of the NMR spin...

Lecture 5: First principle calculation of the NMR spin Hamiltonian parameters including nuclear Zeeman, Shielding tensor and Spin-Spin coupling A) The Nuclear Magnetic Resonance (NMR) spectrum of the helium atom 1. GENERALITY The H atom, and all other paramagnetic molecules as well present peculiar problems essentially due to the fast relaxation of the electronic spin. As a consequence a broadening of the absorption lines results and the NMR signal can no more be observed. Hence we turn to the He atom which is diamagnetic and more appropriate to introduce the basic principle of NMR. The electron pair of He has its spins coupled to form a singlet without magnetic moment et only the nuclear magnetic moment has to be considered. The nuclide 4He has no nuclear spin solely the 3He with an abundance of 0.0001% has to be considered. The nuclear spin possesses 2 states: |α> and |β> in a magnetic field the absorption of energy by the bare nucleus occurs at the resonance frequency : ωN = γNH. La probability P of a NMR transition can be calculated in the same way as for the spin of the electron i.e. the oscillating electromagnetic wave 2H1cos(ωt) produces a perturbation: v(t) = 2γNhH1Ixcos(ωt) from which one obtains P = 2πγN2H12|<αN|Ix|βN>|2g(ω) = πγN2H12|<αN|I++I- |βN>|2g(ω) = (1/2)πγN2H12g(ω) The main feature of this equation is that γN is much smaller than the gyromagnetic factor γe of the electron. Thus, the probability of NMR transitions are roughly 10-5 smaller than those for EPR spectra. 2. CHEMICAL SHIFT OR SHIELDING EFFECT The observed NMR frequencies for 3He as well as for all other nuclei differs slightly from the theoretical value gNβN of the free nucleus. This phenomenon is called chemical shift or shielding effect. It originates from the electronic current that the external magnetic field induces in the atoms. These electric currents induces et the place of the nucleus an additional magnetic field that is opposed to the external magnetic field and whose intensity is proportional to the applied field strength. Thus, the total effective magnetic filed acting onto the nuclear magnetic moment is: Heff = (1-σ)H

where σ is a small number of the order of 10-6 called shielding constant. The expression for the nuclear Zeeman energy has to be modified as: HZe = -gNβN(1-σ)HIz Thus, in NMR spectroscopy a transition at constant frequency will appear at a higher field as for the free nucleus. The theory of the chemical shift in a closed shell atom is quite. In the external magnetic field, the entire spherical electron cloud precesses about the field direction as if it were a rotating rigid sphere of electricity, and its angular velocity is ! = eH

2mc . This is illustrated in the fig. below:

! = eH/(2mc)

v

r

H '

H

Since an electron at a distance r from the nucleus moves with the velocity v = !"r it produces a secondary magnetic field:

H' = - ec r!v

r3 = - e2

2mc2 r!H!r

r3

On average the electrons are distributed over the atom with a probability density ρ(r) and the average secondary field is:

H' = - e2

2mc2

r!H!r

r3 "(r)dv

or

H' = r!j(r)

r3 dv

when the current density j(r) = -e

cv! is known.

When H || z one can show that the components x and y of H' vanish, leaving a screening field -σH directed along the z axis. We evaluate the vector product in the equation below to obtain:

! = e2

2mc2

x2+y2

r3 "(r)dv = e2

3mc2

"(r)

r dv

This relation is also called the Lamb formula, and we see that the screening constant σ depends on the mean value <1/r> for all electrons. For He the integral above works out at 3.377 in units of 1/a0 and the calculated screening constant is σ = 59.93 ppm. B) NMR Spectra of molecules in solution There are two types of interaction which are important: the nuclear Zeeman interaction and nuclear spin-spin coupling. As we saw in the previous section, the surrounding electrons in a molecule produce shielding effects which change the Zeeman term, from H = -gNβNHIz to -gNβN(1-σ)HIz. This is because the actual field at the nucleus is the sum of two magnetic fields, H the applied field, and R' = -σH, the local field at the nucleus arising from the induced electronic currents. H' varies according to the chemical environment of the proton, and if measurements are made with a fixed frequency, the applied field necessary to satisfy the resonance condition also varies from one type of proton to another. The strongest coupling between two nuclear moments is the dipole-dipole coupling discussed in the next section. However we noted that the dipolar interaction tensor is traceless and hence vanishes for a rotating molecule in the liquid phase. Fortunately there are other mechanisms which couple the nuclear spins and although the magnitudes of the couplings are extremely small, they do not vanish for molecules in solution. The most important mechanism leading to correlation of nuclear spins involves polarization of the intervening electron spins. We will see in the next chapter how this occurs. Before embarking on a detailed discussion of the two main interactions, let us consider the familiar example of acetaldehyde, CH3CHO, which illustrates the most important features. First we note the presence of two chemically distinct types of protons, so that the proton resonance spectrum (Fig. below) shows two groups of lines with different relative intensities. The stronger group arises from the three equivalent methyl protons, and the weaker group front the a1dehyde proton. Under conditions of normal resolution these two groups reveal the presence of further structure. The methyl line is split into two components of equal intensity, corresponding to the two allowed orientations of the aldehyde proton spin. The aldehyde proton line shows a more complex splitting into font lines.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50 60

CH3CHO

! = 45.6Hz

J=2.9Hz

3J = 8.7Hz

The separation between the methyl and aldehydic peaks is called the chemical shift; the fine structure within each peak is called the spin-spin splitting. To account for the splitting it la necessary to postulate an isotropic coupling Hamiltonian of the type H = J I1! I2 between each pair of proton spins in the molecule and then analyze the energy levels and transitions within the spin system by the methods used in the previous section. J is called the coupling constant and is measured in cycles pet second (cps). As usual, molecules in a liquid rotate rapidly and J is actually the average over all orientations of an anisotropic spin-spin coupling tensor. 1. THE CHEMICAL SHIFT In liquids the screening tensor σ reduces to an isotropic screening constant σ and the nuclear Zeeman energy becomes

!

ˆ H 0 = "gN#N 1"$( )Hˆ I z In principle the spectrum is obtained by using a fixed frequency ν0, scanning the magnetic field, and determining the value of H required for each proton. If H0 is the resonance field for the bare proton and H, is the field required in a molecule, we have

!

h"0 = #gN$NH0 = #gN$N 1#%( )H1 or

!

" =H1 #H0

H1

$H1 #H0

H0

Thus a positive value of a shifts the resonance to high field. In practice the measurements are made relative to a for some standard compound in the same solvent rather than the bare proton. For instance if the nucleus contains several chemically different protons the Hamiltonian becomes

H = -gN!NH 1-"i Izi!i

and the relative chemical shift of protons 1 and 2 is

!

"1 #"2 =H1 #H2

H0

The chemical shift is often quoted in frequency units by converting the magnetic fields in the equation above into resonance frequencies. The relative shift is then denoted by the symbol δ. δ1 - δ2 = gNβN(H1-H2)/h = (σ1-σ2)gNβNH0/h = (σ1-σ2)ν0 2. SPIN-SPIN COUPLING The acetaldehyde spectrum is a straightforward example of spin-spin coupling, and we shall now interpret it in more detail. The first stage is to write down a suitable spin Hamiltonian. It is most convenient to work in frequency units, so that the Zeeman energy is H0 = - !0 1-"A Iz1 + Iz2 + Iz3 - !0 1-"B Iz4 Here I1, I2, I3 are the methyl proton spins, shielding constant σA and I4 is the aldehyde proton with shielding σB. The spin I4 must couple equally on the average with the three methyl spins, giving a term HAB = J I1 + I2 + I3 ! I4 Couplings will also exist between the methyl protons, but these can be ignored. The important feature which makes a simple analysis possible is that J is much smaller than the relative chemical shift

δ = ν0(σA – σB) of the two kinds of proton. The stationary states of the Zeeman Hamiltonian have the four nuclei quantized along the magnetic field, with quantum numbers m1, m2, m3 and m4 while the zero order energy levels depend only on the total resolved spins of the two groups A and B. Let us define total spin vectors for the groups FA = I1 + I2 + I3 et FB = I4

and new quantum numbers for the operators FzA and FzB MA = (m1 + m2 + m3) and MB = m4 The Zeeman energy now becomes H0 = - !0 1-"A FzA - !0 1-"B FzB The first order spin-spin coupling energy JFA! FB depends only on the diagonal matrix elements of operators like JI1! I4 that is on the value of Iz1Iz4 or m1m4. So we easily see that the total coupling energy is proportional to MAMB. To sum up, the first order energy levels of the spin system are E/h = - !0 1-"A MA - !0 1-"B MB + JMAMB There are two types of allowed n.m.r. transition. In the methyl transitions one of the protons 1,2,3 turns over, so that MA changes by ± 1 and the coupling energy changes by ±JMB. Hence there are two lines at frequencies ν = ν0(1-σA) - JMB (A resonance) ν = ν0(1-σA) ± J/2 The aldehyde proton makes transitions at the frequency ν = ν0(1-σB) - JMA (B resonance) but there are now 4 possible lines corresponding to the different arrangements of methyl spins, i.e.: ααα MA = 3/2 αβα, ααβ, βαα MA = 1/2 ββα, αββ, βαβ MA = -1/2 βββ MA = -3/2 At room temperature all orientations of the spins are equally probable and one observes 2 NMR lines for methyl with equal intensity. However, the resonance of aldehyde consists of 4 equidistant lines separated by J and having intensities in the ratio of 1:3:3:1, which corresponds to the number of methyl spin arrangements. C) Interpretation of the chemical shift and of the spin-spin coupling 1. THE CHIMICAL SHIFT We saw in a previous section that a uniform magnetic field causes the whole electron of an atom to precess about the direction of the of H causing electronic currents to flow. These currents give a secondary magnetic field at the nucleus and lead to the chemical shift.

If the nucleus is part of a molecule, the electrons are, in general, no longer completely free to rotate about the direction of H. This leads to two new effects: (i) that the secondary field H’ is nor necessarily parallel to H and the shielding must be represented by an anisotropic tensor σ; (ii) the theoretical expression for the shielding involves an additional term which often has the opposite sign of the Lamb term. To describe theoretically the chemical shift, let us first consider the classical expression for the current density in absence of an external magnetic field: j = -c

ev!

where v is the velocity of the electron and ρ = ψ∗ψ its density. If we substitute v by pm

(p is the linear momentum of the electron) and represent p by the equivalent quantum mechanical operator, one thus obtains: j = i eh

2!mc "*#" - "#"* (H = 0 )

When ψ represents a closed non-degenerate shell, ψ is real and j = 0 everywhere in the molecule. However, if an external permanent magnetic field is switched on, v has to be represented by v = 1

m p + e

cA

where A = 1

2 r!H is the vector potential of H. This yields

(i) j = i eh

2!mc "*#" - "#"* - e2

mc2 A"*" (H $ 0 )

(ii) a modification of Schrödinger’s equation: - h2

8!2m "

2 + V# + e

mc A$ p + e2

2m2c2A2 # = E# (H % 0)

If the field H is small, the term in A2 can be neglected and the hamiltonian is modified as e

mc A! p = e

2mc H!r " p = e

2mc H" r!p

= eh

4!mc H" L = #H" L

where L h

2! = r"p is the orbital angular momentum.

A rigorous method to calculate the chemical shift is thus: (i) Solve the modified Schrödinger equation as above;

(ii) Calculate the current density

!

r j ;

(iii) Calculate the shielding tensor according to:

H' = -!" H =

r#j

r3 dv

There is another method based on perturbation theory that is generally easier to implement. The perturbed wave function can be obtained as:

! = !0 - "< n | H# L | 0 >

En - E0

!n

excité

where ψ0 represents the wave function in absence of a magnetic field. The secondary field H' that is induced by an electron in

!

r r with a velocity v is expressed by:

H' = - e

c r!v

r3 = - e

mc r! p+e

cA

r3

if we substitute L h by r!p and 1

2H!r for A in the above expression, we obtain:

H' = - eh

4!mc 2L

r3 - e2

2mc2 r"H"r

r3

This classical expression has to be integrated over all space to obtain the average according to the laws of quantum mechanics using the perturbed function ψ, i.e.:

H' = -!" H = - eh

4#mc $ 2L

r3 $ - e2

2mc2 $

r%H%r

r3 $

Finally, substituting in the above expression ψ with the perturbed wave function, we obtain the Ramsey’s formula for the σ tensor which applies to all closed shell molecules; i.e.:

!zz = e2

2mc2 0

x2+y2

r3 0 - eh

4"mc

2

0 Lz n n 2Lz

r3 0

En - E0 +

0 2Lz

r3 n n Lz 0

En - E0!n

excité

and two equivalent expressions for x and y. It is important to note that the chemical shift does not depend whatsoever upon the electron spin! 2. SPIN-SPIN COUPLING

There are in fact many contributions. For the sake of simplicity we only consider here the most important one which is the contact coupling aI! S tending to align the electron spin anti-parallel to the nuclear spin. The other contributions can generally be neglected. Suppose that the nuclear spin of nucleus A is quantized along the z axis, hence the contact interaction can be written as: HA = 8!

3g"gN"NIzASzA

where SzA = ! ri-rA!i

électrons

Szi

Due to this perturbation HA, the wave function of the ground singlet does mix with the excited states and we get:

! = !0 - 8"3

g#gN#NIzAn SzA 0

En - E0 !n!

n

excité

It is thus seen that the wave function depends upon the nuclear spin. Moreover if one considers as well the local polarisation of the electron spin in B i.e.:

SzB = ! ri-rB!i

électrons

Szi

the polarisation interaction of the electron spin in A and B with the nuclear spin gives rise to an interaction of type JSzASzB. The coupling constant J being expressed by:

J = - 8!3

g"gN"NIzA0 SzA n n SzB 0 - 0 SzB n n SzA 0

En - E0!n

excité

D) Perturbation Treatment 1) Nuclear Spin A nucleus with non-zero spin acts as a magnetic dipole, giving raise to a vector potential AA

AA =µ0IA ! r "RA( )4#r "RA

3

Here IA is the magnetic moment of nucleus A and RA is the position (the nucleus is the natural Gauge origin). Adding this to the external vector potential and expanding gives

p +A +AA( )2 = p2 +A2 +AA2+ 2A !p + 2AA !p + 2A !AA

The p !AA , AA

2and A !AA terms give the following operators, respectively.

HPSO =µ04!

IA " r #RA( ) $ p[ ]r #RA

3= IA "P1

PSO

HDSO

=µ02

32!2

IA $ r #RA( )[ ] " IB $ r #RB( )[ ]r #RA

3r# RB

3= IA "P2

PSO"IB

H% =µ08!

B $ r #RG( )[ ] " IA $ r #RA( )[ ]r #RA

3= B "P2

% " IA

HPSO and HDSO are called the Paramagnetic and Diamagnetic Spin-Orbit operators, whileHσ gives the, diamagnetic part of the nuclear shielding. HPSO depends only on one perturbation (nuclear spin) and is a P1 operator in the nomenclature above, while HDSO and Hσ refer to two perturbations, and are of the P2 type. They may be written as

P1PSO =

µ04!

r "RA( ) # p

r "RA3

P2DSO

=µ02

32!2r "RA( )t r "RB( ) " r "RA( ) r" RB( )t

r" RA3r "RB

3

P2$ =

µ08!

r "RG( )tr "RA( ) " r "RG( ) r "RA( )t

r "RA3

where the notation for P2

DSO and P2! is analogous to that in chap. 6 and RA/B denotes nuclear positions.

The NMR shielding, which is the mixed second derivative with respect to, a nuclear spin and an external magnetic field, has in analogy with the magnetizability a diamagnetic and a paramagnetic part. The diamagnetic part arises from P2

! while the paramagnetic contribution contains products of matrix elements from P1

PSO (from the nuclear spin) and the angular moment operator L (from the external field). Written in terms of the perturbation formula (18), the expression for the nuclear shielding for atom A becomes

!A= "0 P2

!"0 +

"0 P1PSO "i "i L "0 + "0 L "i "i P1

PSO "0

E0 # Eii =1

$

%

All the operators P2

! , P1PSO and L are gauge dependent, relating to the position of atom A and each of

the dia- and para-magnetic terms depends on the chosen gauge. In order to describe nuclear spin-spin coupling, we need to include electron and nuclear spins, which are not present in the non-relativistic Hamilton operator. A relativistic treatment, gives a direct nuclear-nuclear coupling term:

HnnSS

=µ08!

IA " IBRA #RB

3# 3

IA " RA #RB( )[ ] RA #RB( ) " IB[ ]RA #RB

5

$

%

& &

'

(

) )

For rapidly tumbling molecules (solution or gas phase) this contribution averages out to zero, but it is significant for solid state NMR. The Hnn

SS operator gives an indirect term, which is normally written as two separate operators: HFC

=geµ03

s !IA( )" r #RA( ) = IA !P1FC

HSD

= #geµ08$

s ! IA

RA #RB3 # 3

s ! r #Ra( )[ ] RA #RA( ) ! IA[ ]RA #RB

5

%

&

' '

(

)

* * = IA !P1

SD

HFC is the Fermi Contact and HSD is the Spin-Dipolar operator, where s is the (electron) spin

operator. HFC and HSD depend only on one nucleus and are thus P1 type operators.

!

P1FC =

geµ03

" r #RA( )s

P1SD = #

geµ08$

s

RA #RB3# 3

s % r #Ra( )[ ] RA #RA( )

RA #RB5

&

'

( (

)

*

+ +

The HFC and HSD operators determine the isotropic and anisotropic parts of the hyperfine coupling constant (eq. 11), respectively. The latter contribution averages out for rapidly tumbling molecules (solution, or gas phase), and the (isotropic) hyperfine coupling constant is therefore determined by the Fermi-Contact contribution, i.e. the electron density at the nucleus. The indirect spin-spin coupling between nuclei A and B which is the one observed in solution phase NMR, contains several contributions

!

JAB = "0 P2,ABDSO "0 +

"0 P1,A "i "i P1,B "0

E0 # Eii$0

%

The first part can be evaluated as the expectation value of P2

DSO . The second part contains six pieces, corresponding to all combinations of P1

PSO , P1FC and P1

SD . P1FC and P1

SD contain the electron spin operators and for a singlet ground state (as is usually the case), this means that the excited state ψi in the summation must be a triplet state. Since P1

PSO does not depend on electronic spin, this means that the combinations of P1

PSO with either P1FC or P1

SD give zero contribution. For rapidly tumbling molecules, it can be shown the cross term between P1

FC and P1SD averages out. For the trace (sum of

the diagonal terms) of the 3 x 3 coupling matrix J which is the observed coupling constant, only the three "diagonal" terms (Pl = Pl’) in thus survive. It is often found that the Fermi-Contact term is the most important, especially for one-bond couplings (2J), followed by the Paramagnetic Spin-Orbit, while the Diamagnetic Spin-Orbit and Spin-Dipolar contributions are small. It should be noted, however, that this is based on relatively few data.

2. Gauge Dependence of Magnetic Properties There are two factors which make the calculation of magnetic properties somewhat more complicated than. the corresponding electric properties. First, the angular momentum operator L is imaginary, implying that the wave function must be allowed to be complex. Second, the presence of the gauge origin in the, operators means that the results may be origin-dependent. An exact wave function will of course give origin dependent results, as will a HF wave function if a complete basis set is employed. In practice, however, a finite basis must be employed, and standard basis sets will yield results which, depend on where the user has chosen the origin of the gauge to be. The centre of mass is often used, but this is by no means a unique choice. The gauge error depends on the distance between the wave function and the gauge origin, and some methods try to minimize the error by selecting separate gauges for each (localized) molecular orbital. Two such methods are known as Individual Gauge for Localized Orbitals (IGLO) and Localized Orbital/local oRiGin (LORG). A more recent implementation, which completely, eliminates the gauge dependence, is to make the basis functions explicitly dependent on the magnetic field by inclusion of a complex phase factor referring to the position of the basis function (usually the nucleus).

!A r,RA( ) = e"i

cAA #r

$A r,RA( )

$A r,RA( ) = Ne"% r"RA( )2

AA =1

2B & RA "RG( )

Such orbitals are known as London Atomic Orbitals (LAO) or Gauge Including/Invariant Atomic Orbitals (GIAO). The effect is that matrix elements involving GIAOs only contain a difference in vector potentials, thereby removing the reference to an absolute gauge origin. For the overlap and potential energy it is straightforward to see that matrix elements become independent of the gauge origin.

!A !B = "A e

i

cAA #AB( )$r

"B

!A V !B = "A e

i

cAA #AB( )$r

V "B

AA #AB =1

2B % RA #RB( )

The kinetic energy is slightly more complicated, but it can be shown that the following relation holds:

!A "2 !B = !A p +1

2B # r $RG( )

%

& ' (

) * 2

!B

= +A e

i

cAA$A

B( ),rp +

1

2B # r $RB( )

%

& ' (

) * 2

+B

Note that RG has been replaced by RB in the last bracket. The use of GIAOs as basis functions makes all matrix elements, and therefore all properties, independent of the gauge origin. The wave function itself, however, is expressed in term of the basis functions, and therefore becomes origin dependent, by means of a complex phase factor. The use of perturbation-dependent basis functions has the further advantage of strongly reducing the need for high angular momentum functions, i.e. the property is typically calculated with an accuracy comparable to that of the unperturbed system. While LAO/GIAO had been proposed well before the advent of modern computational chémistry, it was only developments in calculating (geometrical) derivatives of the energy (and wave function) that made it practical, to use field-dependent orbitals.

NMR

Among the calculation of second-order magnetic response properties the calculation of nuclear magnetic shielding tensor plays a relevant role. It's generally known that the calculation of such a property requires gauge invariant procedures. The GIAO (Gauge Including Atomic Orbital)[1] and the CSGT (Continuous Set of Gauge Transformations)[2] are available in the standard Gaussian94 package.

General Procedure

In order to get the relative shift in ppm it is necessary to know the absolute shitfs of the nuclei of the analysed molecule and those of the standard (SiMe4, TMS).

See an Example of Input for a Standard Calculation

Exercise 1: 13C and 1H NMR Spectra of 1,10-phenantroline

(in vacuum)

I. Compute the Equilibrium Structure

a. Recommended starting Geometry: dCC = 1.42 dCH = 1.01 dCN = 1.32 Symmetry C2v

b. Optimise the Structure: B3LYP/6-31G* II. On the Optimised Structure

a. Compute the Absolute NMR Shielding Tensor with GIAO Method with the following Basis Sets

I. B3LYP / 6-31G* II. B3LYP / 6-311+G(2d,p)

III. HF / 6-31G* IV. HF / 6-311+G(2d,p)

b. Compute the Absolute NMR Shielding Tensor with CSGT Method with the following Basis Sets

. B3LYP / 6-31G* I. B3LYP / 6-311+G(2d,p)

II. HF / 6-31G* III. HF / 6-311+G(2d,p)

III. By difference with TMS Absolute Shielding Values (Table I.) compute the relative Shifts.

IV. Compare with the Experimental Values given in Table II.


See the Optimised Geometry of Phenantroline

See an Example of Output

Questions:

What is the Effect of the Basis Set and of the Method Used on the relative Shift?

Results

Exercise 2: 1H and 13C NMR Spectra of Uracil.

(in vacuum)

I. Compute the Equilibrium Structure

a. Recommended starting Geometry: Symmetry Cs

b. Optimise the Structure: B3LYP/6-31G* II. On the Optimised Structure

a. Compute the Absolute NMR Shielding Tensor using CSGT Method and : I. B3LYP / 6-31G*

II. B3LYP / 6-311+G(2d,p) III. HF / 6-31G*

IV. HF / 6-311+G(2d,p) III. By difference with TMS Absolute Shielding Values (Table III.) compute the

relative Shifts. IV. Compare with the Experimental Values given in Table IV.


See the Optimised Geometry of Uracil

Question:

What is the Effect of the Basis set on the relative Shifts ?

Results

Table I

Computed Absolute Isotropic Shielding Values for TMS in vacuum

GIAO 13C 1H

B3LYP / 6-31G* 189.6621 32.1833

B3LYP / 6-311+G(2d,p) 182.4485 31.8201

HF / 6-31G* 199.9711 32.5957

HF / 6-311+G(2d,p) 192.5828 32.0710

CSGT 13C 1H

B3LYP / 6-31G* 188.5603 29.1952

B3LYP / 6-311+G(2d,p) 182.1386 31.7788

HF / 6-31G* 196.8670 29.5517

HF / 6-311+G(2d,p) 192.5701 31.5989

back

Table II

Experimental Shifts Relative to TMS for 1,10-phenantroline (in ppm). a

13C

1H

149.7; 145.4; 135.3; 128.1; 126.4; 123.0

9.13; 7.64; 8.28; 7.82

a T. A. Annan, R. K. Chaudha, D. G. Tuck, K. D. Watson, Can. J. Chem. 65 (1987) 2670.

Back to Exercise 1

TOP

Table III

Computed Absolute Isotropic Shielding Values for TMS in Vacuum using the CSGT Method

CSGT 13C 1H

B3LYP / 6-31G* 189.6621 32.1833

B3LYP / 6-311+G(2d,p) 182.1386 31.7788

HF / 6-31G* 196.8670 29.5517

HF / 6-311+G(2d,p) 192.5701 31.5989

Table IV

Experimental Shifts Relative to TMS for Uracil (in ppm).

C2a

C4a C5

a C6a H1

b H3b H5

b H6b

151.6 171.2 100.2 147.3 10.90 11.08 5.49 7.44

a in solid from S. Ganapathy et al., J. Am. Chem. Soc. 103(1981) 6011.

b in DMSO-d6 from R. H. Griffey et al., J. Am. Chem. Soc. 107(1985) 711.

Back to Exercise 2

TOP

References:

[1] R. Ditchfield, Mol. Phys. 27, 789 (1974)

[2] T. A. Keith, R. F. W. Bader, Chem. Phys. Lett. 210, 223 (1993)

Results for Exercise 1

Absolute Shifts for 13C and 1H of 1,10-phenantroline

GIAO B3LYP / 6-31G* B3LYP / 6-311+G(2d,p) HF / 6-31G HF / 6-311+G(2d,p)

C1 46.5971 26.6743 48.7964 33.4740

C2 74.2507 56.1533 80.6274 67.1479

C3 61.5028 42.7073 63.1065 48.1134

C4 68.5223 50.6044 75.8289 62.3665

C5 66.9860 48.0139 73.1853 59.2079

C6 47.5352 27.2707 50.2081 34.9139

H1 22.9415 22.2692 22.93228 22.4659

H2 24.8343 24.1012 24.9235 24.2526

H3 24.3362 23.4689 24.1134 23.3533

H4 24.6405 23.8338 24.7166 23.9912

CSGT B3LYP / 6-31G* B3LYP / 6-311+G(2d,p) HF / 6-31G HF / 6-311+G(2d,p)

C1 42.4461 26.8165 42.4046 33.9082

C2 71.5528 56.4698 75.2335 67.7271

C3 57.5795 42.6698 57.1321 48.3597

C4 64.4355 50.6151 69.1971 62.5560

C5 64.1489 48.6269 68.4587 59.9022

C6 42.6814 28.1051 44.3695 35.8919

H1 22.9158 22.2661 23.1862 22.3959

H2 24.4770 24.0461 24.6261 24.1256

H3 24.0437 23.3576 24.0742 23.1736

H4 24.3172 23.7580 24.5674 23.8484

Back to Exercise 1

Results for Exercise 2

Relative (to TMS) Shifts in ppm for 13C and 1H of Uracyl using the CSGT Method

Vacuum

B3LYP / 6-31G* HF / 6-31G* B3LYP / 6-311+G(2d,p) HF / 6-311+G(2d,p)

C2 143.8605 154.9303 153.7714 159.7213

C4 156.3871 166.4363 165.5726 171.7239

C5 107.9558 101.3658 106.0694 100.9926

C6 134.6589 146.004 143.4008 150.4223

H1 2.4371 2.3562 6.2275 5.7942

H3 2.8121 2.3522 7.0661 6.8216

H5 2.3034 2.4017 5.5277 5.1223

H6 3.7936 4.7591 6.9862 6.8845

Back to Exercise 2

LFDFT: NUCLEAR MAGNETIC RESONANCE Nuclear magnetic resonance is magnetic resonance using the nuclear spin

!

r I and magnetic moment

!

r µ = "

N#

N

r I .

!

r I refers to the lowest nuclear level and is the total angular momentum vector for the

nucleus. It is customary to do nuclear resonance on compounds which have singlet electronic ground states and we discuss this case for hexa-coordinated d6, low-spin, transition metal ions. The direct interaction of the nuclear magnetic moment with an external magnetic field

!

r H is

!

"r µ #

r H .

However, the actual interaction is different because the field at the nucleus is modified by the magnetic moment induced in the electronic system by the external field. It is conventional to write the actual first-order interaction with

!

r H as

!

" 1"#( )r µ $

r H . So

!

"r µ #

r H represents the modification and σ is called

the chemical screening constant. σ has two parts. One originates from the nuclear hyperfine interaction described in the appendix and the other from the diamagnetic part of the Hamiltonian for the interaction of the electronic system with magnetic fields. We now evaluate these in order. To start, we know that the ground state for a hexa-coordinated d6, low-spin complex in a magnetic field

!

r H , corresponds to the eigenstate with lowest energy of the following Hamiltonian within the d6 ligandfield manifold:

!

ˆ " = ˆ h icore

i

# + VLF +1

riji< j

# + $ nd

r l i %

r s i

i

# + & kr L + ge

r S ( ) %

r H

= ˆ H 0

+ &r Z %

r H

where:

!

ˆ h i

core is the one-particle core Hamiltonian for d-orbitals on the transition metal ion,

!

VLF

is the ligandfield potential obtained in a LF-DFT calculation [vide supra] as

!

di VLF d j matrix elements,

!

1

rij represents the electrostatic repulsion between pairs of d electrons whose matrix elements are

also obtained in a LF-DFT calculation [ref] in term of Racah’s parameters,

!

"nl

= Rnl

1

r

dV

drRnl

obtained from the radial part of the |ndi> orbitals,

k is an orbital reduction factor equal to the averaged |ndi> population, ge= 2.00232. For moderate magnetic fields as those used in NMR spectroscopy, the Zeeman term in the Hamiltonian expression above is always much smaller than

!

ˆ H 0. Thus, perturbation theory can be employed to

express the 1st order magnetic field dependence of the ground state wave function. That is, we solve

first the secular equation:

!

"µˆ H

0"# where

!

"µ are the

!

10

6

"

# $

%

& ' = 210 possible single determinants

!

"11( ) ... "6 6( ) that can be obtained by occupying the 10 d spinorbitals

!

" i =#

dxy ,+

dxy ,#

dyz ,+

dyz ,#

dz2 ,

+

dz2 ,

#

dxz ,+

dxz ,#

dx2#y 2 ,

+

dx2#y 2

$ % &

' ( )

with 6 electrons. Diagonalisation of

this 210x210 matrix yields the eigenvalues εi and eigenfunctions

!

"i= c

iµ#µµ=1

210

$ of

!

ˆ H 0. Let us denote by

ε0 and ψ0 the eigenvalue and eigenfunction of lowest energy respectively. The ground state

!

0 correct to 1st order in the magnetic field

!

r H is then

!

0 = "0

+ #"n

r Z $

r H "

0

%n &%0"n

n'0

( = "0

+ # H)

"n Z) "0

%n &%0"n

n'0

()= x,y,z

(

= "0

+ # H)

"n kL) + geS) "0

%n &%0"n

n'0

()= x,y,z

(

where Hα are the spatial components of the external magnetic field. Thus, to first order in

!

r H , we find

that the induced coupling to the nuclear moment is

!

0 ˆ H hyperfine 0 = P 0r " # I 0

= P $0

+ % H&

$m Z& $0

'm ('0

$mm)0

*&= x,y,z

*r " # I $

0+ % H& '

$n Z& '$

0

'n ('0

$nn)0

*& '= x,y,z

*

= P $0

r " # I $

0+ 2P% $

0

r " # I H& '

$n Z& '$

0

'n ('0

$nn)0

*& '= x,y,z

* + ' H2( )

+ P $0

r " # I $

0+ 2P% I&

$0"& $n $n Z& '

$0

'n ('0n)0

*,

- .

/

0 1

&= x,y,z& '= x,y,z

* H& '

Using

!

P = ge"N#e#N r$3 and keeping in mind that the contribution to the chemical screening constant

is the coefficient of

!

r µ "

r H we get

!

"## 'p

= 2ge$e

2r%3 &

0'# &n &n Z# ' &0

(n %(0n)0

*

where σp is the paramagnetic contribution to the chemical screening tensor σ defined as:

!

ˆ H nuclear _ Zeeman

=r µ t (1-σ)

!

r H

In order to illustrate this finding, let us consider the simple case of octahedral low-spin cobalt(iii) compounds. In this case, the |t2

6 1A1> ground state is not much affected by spin-orbit coupling and the screening tensor will be isotropic because of the octahedral symmetry. Thus, only the z-component of the Zeeman and of the hyperfine operators has to be considered. Moreover, among the excited ligandfield states only |t2

5 e1 1T1> has a non-zero matrix element with the ground through the Zeeman operator

!

t2

6 1A1Zz t2

5e1 1T1

= t2

6 1A1kLz + geSz t2

5e1 1T1

= k t2

6 1A1Lz t2

5e1 1T1

= "2ki 2

As both the ground and excited terms are singlets, the only part of the nuclear hyperfine interaction Ωz which contributes is Lz. If we denote by E1 the energy separation between

!

t2

6 1A1

and

!

t2

5e1 1T1

, the paramagnetic screening constant for cobaltic compounds reads:

!

" p =" zz

p = 2ge#e

2r$3 %

0&z %n %n Zz %0

'n $'0n(0

) = 2ge#e

2r$3

t2

6 1A1&z t2

5e1 1T1t2

5e1 1T1Zz t2

6 1A1

E1

= 2ge#e

2r$3

$2ki 2( ) 2i 2( )E1

= $16k ge #e

2r$3

E1

To obtain the diamagnetic contribution we go back to equation:

!

2mc"

e= 2 f

r A #

r p + fh

r $ # rot

r A +

r $ #

r p f( )

r $ #

r A ( ) +

e

cfr A 2

of the appendix, with f=1. The diamagnetic term in the Hamiltonian for the electronic system is

(e2/2mc2)A2, where A is the sum of the vector potential

!

r A 1

=1

2

r H "

r r for the static external field and

!

r A 2

=

r µ "

r r

r3

for the magnetic field due to the nuclear moment. It is only the cross-terms in (A1+A2)2

which give coupling between the nuclear moment and the external field. Hence the coupling is

!

e2

mc2

r A 1"

r A 2

=e2

2mc2r3

r H #

r r ( ) "

r µ #

r r ( )

=e2

2mc2r3

r2r H "

r µ $

r H "

r r ( )

r µ "

r r ( )[ ]

Now if we evaluate this expression over the 1A1 ground term only those parts which belong to the spatial representation A1 contribute. In

!

r H "

r r ( )

r µ "

r r ( )we have a sum of products like Hxµxx2 and Hxµyxy.

The only combination of x2, xy, etc., which transforms as A1 is x2+y2+z2 = r2 and hence the mean values over angle satisfy

!

xy = yz = zx = 0, x2

= y2

= z2

=1

3r2

Introducing this finding our expression simplifies to

!

e2

2mc2r3

r2r H "

r µ #

r H "

r r ( )

r µ "

r r ( )[ ] =

e2

2mc2r3

r2r H "

r µ #

1

3r2r H "

r µ

$

% & '

( ) =

e2

3mc2r

r H "

r µ

giving a “diamagnetic” contribution

!

" d=

e2

3mc2r#1

=e2

3mc2r#1

=1

3$ 2e2r#1

to the chemical screening constant and where α = 1/137.037 is the fine structure constant If we now take the Z axis along the direction of the external field, the energy levels of the nucleus are at -(1-σ)γβNHMI and for absorption and emission of electromagnetic radiation we have the usual selection rule ΔMI = ±1. Therefore we may expect one line at a frequency ν satisfying:

hν = (1-σ) γβNH,

where

!

" =" p+" d

= #16kge$e

2

E1

r#3

+1

3% 2e2r#1

Assuming that

!

r"3 and

!

r"1 remain the same for all cobaltic compounds we deduce that for a

constant external field H there is a linear relation between ν and the wave-length of the lowest 1A1-1T1 transition

Appendix: The hyperfine interaction of an electron with the nuclear magnetic moment This development was already given in chap. 2 and is repeated here for the sake of completness. Here we take

!

A0

=Ze

r

and

!

r A =

"#N

h

r I $

r r

r3

where I is the angular momentum vector for the nucleus,

!

"N

=eh

2Mc is the nuclear magneton and γ is a

pure number: the gyromagnetic factor of the nucleus. γ = 5.586 for the proton and would be γ = 2 if Dirac’s equation applied to the proton. The reason for the deviation is not known. The vector I

commutes with the other observables occurring in the problem. It is convenient to write

!

r µ =

"#N

r I

h for

the magnetic moment of the nucleus. We obtain then the classical formula

!

r A =

r µ "

r r

r3

for the vector potential due to a dipole situated at the origin, but one must not forget that the components of µ do not commute with one another. If we write Δψ2 for the additional term in Dirac’s equation due to this interaction, we get.

!

" =1

2m

r # $

r p +

e

c

r A

%

& '

(

) * f

r # $

r p +

e

c

r A

%

& '

(

) * +

r # $

r p f

r # $

r p

, - .

/ 0 1

and so

!

2m" =e

c

r # $

r A ( ) f

r # $

r p ( ) +

e

c

r # $

r p ( ) f

r # $

r A ( ) +

e2

c2

r # $

r A ( ) f

r # $

r A ( )

The factor f is reminiscent of the elimination of the small component in Dirac’s equation, i.e.

!

f =2mc

2

2mc2

+ E + eA0

"1#E + eA

0

2mc2

. Recall basic electrodynamics and vector analysis:

!

rot

r A = "

r µ

r3

+3

r µ #

r r ( )

r r

r5

; div

r A = 0

Let us use these relations to transform the equation above:

!

2mc"

e= f

r A #

r p + i

r $ #

r A %

r p ( ) +

r $ #

r p f( )

r $ #

r A ( ) + f

r p #

r A + i

r $ #

r p %

r A ( ) +

e

cfr A 2

= 2 fr A #

r p + fh

r $ # rot

r A +

r $ #

r p f( )

r $ #

r A ( ) +

e

cfr A 2

We now transform the 1st three terms further:

!

2 fr A "

r p =

2 f

r3

r µ #

r r "

r p =

2 f

r3

r µ "

r r #

r p =

2 f

r3

r µ "

r l

!

fhr " # rot

r A = fh $

r µ #

r "

r3

+

r µ #

r r ( )

r " #

r r ( )

r5

% & '

( ) *

!

r " #

r p f( )

r " #

r A ( ) = $

ih

r

df

dr

r " #

r r ( )

r " #

r A ( )

= $ih

r

df

dr

r r #

r A + i

r " #

r r %

r A ( ) =

h

r4

df

dr

r " #

r r %

r µ #

r r ( ){ }

=h

r2

df

dr

r µ #

r " $

r µ #

r r ( )

r " #

r r ( )r$2{ }

On substituting these findings in the original equation we obtain:

!

" =v B #

r µ +

e2

2mc2

f

r µ $

r r ( )

2

r6

where:

!

v B =

e

mcf r

"3r l " r

"3r s + 3r

"5 r s #

r r ( )

r r [ ] + r

"2 df

dr

r s " r

"2 r s #

r r ( )

r r [ ]

$ % &

' ( )

The 2nd term in the equation above belongs to the next order in a perturbation expansion in powers of µ and we therefore reject it. We must now evaluate

!

" = #$Nh%1

r B &

r I

for an atomic state. B commutes with I and is a tensor operator of rank 1 with respect to the electronic angular momentum j. The 2nd factor I commutes with j and is a tensor operator of rank 1 with respect to I. Writing f = j + I for the total angular momentum we deduce that Δ commutes with f. We wish to know the matrix elements of Δ within the set of states |jIfmf> which arise from a given level of the atom and, of course, from the ground state of the nucleus. f ranges from |I-j| to I+j. The commutation rules listed above will enable us to simplify the calculation of the energy eigenvalues in the same way as for the spin orbit coupling in deriving the Landé interval rule. We have

!

E f( ) = jIfmf " jIfm f =1

2a f f +1( ) # I I +1( ) # j j +1( )[ ]

where a is the hyperfine coupling constant. For a one-electron atom with

!

j = l ±1 2 we have

!

a = 2"#e#N r$3 l l +1( )j j +1( )

.

In order to facilitate the calculation it is useful to replace parts of Δ by equivalent operators (Wigner-Eckart theorem). It is convenient to do this in stages and we take the 1st bracket first and let f=1. Then the z-component of Δ is

!

"r"3sz + 3r"5 xzsx + yzsy + z2sz( )= 3r"5 xzsx + yzsy +1

33z

2 " r2( )sz#

$ % &

' (

Remember now that if we have to determine matrix elements of a rank two tensor operator (2z2-x2-y2, x2-y2, xy, xz, yz) we 1st express them as symmetrical quadratic form, for example : xy = (xy+yx)/2 and then replace each of the x, y, z by the corresponding component of J. That is :

!

2z2" x

2" y

2 =# 2Jz2" Jx

2" Jy

2( )x2" y

2 =# Jx2" Jy

2( )

xy =1

2# JxJy + JxJy( )

xz =1

2# JxJz + JzJx( )

yz =1

2# JyJz + JzJy( )

Thus, an operator equivalent to the term above is

!

"3# r"31

2lxlz + lzlx( )sx +

1

2lylz + lzly( )sy +

1

32lz

2 " lx2 " ly

2( )sz$

% & '

( )

where ξ is a dimensionless constant. This expression is then rearranged to read:

!

" r#3

l l +1( )sz#3

2

r l $

r s ( )lz

#3

2lz

r l $

r s ( )

%

& ' (

) *

Thus, the first three terms in the Δ-expression may be replaced by :

!

2"#e#

Nr$3

r l $ r

$3r s + 3r$5

r s %

r r ( )

r r [ ] %

r I = 2"#

e#

Nr$3

r l %

r I + & l l +1( )

r s %

r I $

3

2

r l %

r s ( )

r l %

r I ( ) $

3

2

r l %

r I ( )

r l %

r s ( )

'

( ) *

+ , - . /

0 1 2

with

!

" =2

2l #1( ) 2l +1( ). For d-electrons l = 2, so ξ = 2/21.

The last part in the Δ-expression may be replaced by an expression due to Fermi involving a delta-function (cf. Griffith for details):

!

2"#e#N r$2 df

dr

r s %

r I $ r

$2 r s %

r r ( )

r r %

r I ( )[ ] & 1

316'( )"#e#N( rk( )

r s %

r I

Finally, the complete hyperfine hamiltonian reads:

!

" hyperfine = 2#$e$N r%3

r l &

r I + ' r

%3l l +1( )

r s &

r I %

3

2

r l &

r s ( )

r l &

r I ( ) %

3

2

r l &

r I ( )

r l &

r s ( )

(

) * +

, - +8.

3/ rk( )

r s &

r I

0 1 2

3 4 5

The first term corresponds to the interaction of the nuclear spin with the orbital angular momentum of the electron, the second to the interaction of the nuclear spin with the electronic spin and the last term, involving the delta-function, is called the Fermi contact term. Finally the n-electron hyperfine Hamiltonian for a dn system may be taken as:

!

ˆ H hyperfine = Pr L "

r I #$

r S "

r I +

1

7

r a k "

r I

k=1

n

%&

' (

)

* + = P

r , "

r I

where:

!

P = ge"N#e#N r$3 ,

!

r a

k= 4

r s

k"

r l k#r s

k( )r l k"

r l k

r l k#r s

k( ) ,

!

"P =8#

3$% 0( ) & $' 0( )[ ] and

!

r " =

r L #$

r S +

1

7

r a

k

k=1

n

%

Assessment of theoretical prediction of the NMR shielding tensor of 195PtClxBr6-x

2- complexes by DFT calculations: experimental and computational results.

Peter Belser1, Henry Chermette2, Claude Daul1, Emmanuel Penka1

1Département de Chimie, Université de Fribourg, chemin du Musée 9, CH-1700 Fribourg

2Laboratoire de Chimie Physique Théorique, bât 210, Université Claude Bernard Lyon-1 and UMR CNRS 5182, 43 bd du 11 novembre 1918, 69622-Villeurbanne cedex, France.

Abstract In the present work, the ZORA spin-orbit Hamiltonian, in conjunction with the gauge including orbital (GIAO) method based on DFT theory has been used to calculate 195Pt chemical shift of 195PtClxBr6-x

2- complexes. Excellent agreement with experiments has been obtained for calculations bearing on optimized geometries and all electrons triple zeta + polarization (TZP) STO basis sets: the relative error with respect to experiment amounts to less than 1.5%. It is found that the Pt chemical shift is dominated by the paramagnetic and the spin orbit contribution, whereas the diamagnetic term remains negligible. The influence of the quality of the basis sets have been studied and found small provided a basis set like TZP is used. Several calculations have been performed in order to establish the sensitivity of the chemical shift to a variation in the bond lengths. A strong dependence has been found, with an increase of the chemical shift amounting 150 ppm/pm for a distances decrease. A large sensitivity to the solvation, leading to changes in the structure, is then expected. Different tests using conductor like screening models have been performed in order to establish the sensitivity of the chemical shift to solvation. It has been observed that the changes in the geometry are more important than charge transfers. Finally the sensitivity of the system to the exchange-correlation functional is found rather weak, at least among the GGA functionals. Keywords: NMR, solvation, shielding tensor, relativistic effects, platinum complexes. Introduction The first experimental measurements of the chemical shifts in Pt halides were reported in 1968 by von Zelewskyi. He observed that mixed platinum halides exhibit strong chemical shifts. The same year Dean ii and Green published the first predictions of platinum chemical shift. For the first time Ramsey equationsiii were used for the calculation of the paramagnetic tensor known to be the majority contribution of the shielding tensor in PtX4 planar square complexes, showing that the orbital energy gap contributed to the platinum chemical shift less than did the covalence of the platinum-ligands bonds. These pioneer calculations on platinum have been of limited use for a while, because of the high number of electrons in platinum, involving a large amount of computer time and requiring relativistic corrections to be taken into account. In order to reduce the computational demand of such systems, Kaupp et al.iv have for the first time proposed a DFT approach involving the use of pseudo potentials for the representation of the electronic core of transition metals. Indeed, the break through in the calculation of the NMR chemical shift can be related to the work of Malkinv et al. who were the first to compute this property through a DFT approach with Kohn-Sham independent gauge for localized orbitals

(IGLO)vi. Later a significant development in the computation of the chemical shift has been brought by the works of Schreckenbachviia,b,c where the calculation of NMR shielding tensors using gauge including atomic orbitals (GIAO) and

modern density functional theory has been applied to calculate 17O absolute shielding in transition metal oxides [MO4]n- . This model has also successfully been used by Gilbert to calculate the 195Pt chemical shift for a series of Pt (II) complexesviii. Finally, for sake of completeness, we also mention that Pickart and Mauriix have extended the Blöchlx projector augmented wave (PAW) in order to include the gauge, leading to calculations of the NMR chemical shifts with pseudo potentials. Their gauge-including projector augmented wave (GIPAW) permits to calculate NMR chemical shifts with a pseudopotential approach leading to results comparable to GIAO all electron calculations. Experimental details 195Pt is a nucleus with ½ spin and an isotopic abundance of 33.7% well suitable for NMR experiments. In the practical experimentsxi, measurements of the peak area are performed, corresponding to an aqueous solution of Na2PtCl6 1mol/L concentration in which a solution of sodium bromide 1.5 molar is added. A substitution of chlorine by bromine in the complex follows, leading to different peaks in the spectrum. NMR experiments were performed with a Brucker 300MHz apparatus, at room temperature. The change in the chemical environment around platinum leads to the observation of different peaks with quasi constant intervals as reported in Table III. The complex system will contain some isomers, such as cis and trans [PtCl4Br2]-2), in that case, the abundance related to these isomers (4/5 for the cis form) enables to differentiate them through the observation on the spectrum of the integrals amounting 128.2 and 31.6 for a shift amounting 582-583 ppm, corresponding to the ratio 0.80 and 0.19, respectively, in agreement with the theoretical prediction. Theoretical details Frozen core Approximation. It has been retained for the geometry optimization of the complexes. This approximation assumes that Molecular Orbitals (MOs) describing inner shell electrons remain unperturbed in going from a free atom to a molecule. Thus, these inner electrons can be excluded from the variational procedure, and, instead, they are pre-calculated in an atomic calculation and kept frozen thereafter. The justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than are the valence electrons. On the contrary, for the calculation of the shielding tensor (vide infra), it is necessary to take into account any change in all orbitals, including core orbitals which, being localised near the nuclei, are strongly interacting with the nuclear spin momentum. We have therefore calculated the shielding tensor with all electrons basis sets, and also performed calculations with frozen cores, in order to evaluate the contribution of core orbitals to the evaluation of the chemical shift. Relativistic Zora approximation For heavy atoms, it is well known that an accurate description can only be obtained if additional effort is carried out to solve the Schrödinger equation; knowing that the electrons move fast near an heavy atom nucleus, the effects the theory of special relativity must be taken into account. In 1928, Diracxii proposed the following equation which makes the quantum theory compatible with the relativity theory: Нrψ=Eψ with Hr=cα.p+βc2+V (1)

where α (component: x, y, z) and β are Dirac’s 4x4 matrices, p is the momentum operator , V is the potential, ψ is the wave function with four relativistic components, and H is the Hamiltonian of the system; c is the speed of light, and atomic units are used throughout. Note that in Dirac’s formulation, the particles can have positive or negative energies and the energy spectrum contains the bound states in between a positive and a negative energy continuum. By some standard transformations on the wave function, the explicit form of the Dirac equation leads to the following equation, eliminating the so-called small component of the wave function:

!!""!! Epc

VEpVH

esc=#

$

%&'

( )++=

**)

**

.2

1.2

11

2 (2)

where esc

H denotes the Dirac Hamiltonian (eliminating the small component) for the remaining large component! . By comparison of this equation with the non relativistic Schrödinger one, the kinetic part is replaced by a more complicated operator:

!!"

!!

#$

%&'

( "+! p

c

VEpp .

21.

2

1

2

11

2

2 )) (3)

Like the Schrödinger equation, the Dirac equation cannot be solved exactly for many electron systems. Different types of approximation have been proposed.

The first one, expanding escH in

2

2

c

p , and keeping the zeroth and the first order terms, is known as

Pauli Hamiltonian.

).(4

1

882 22

2

2

42 !!!

"#+#

+$+= pVcc

V

c

ppVH pauli % (4)

where 2

4

8c

p! is the so-called mass velocity term,

2

2

8c

V! is the Darwin term,

).(4

12

!!!

"# pVc$ is the spin-orbit term.

The current problem with this approximation is that in the core region, the assumption that p2<<4c2 is inappropriate for heavy atoms and thus one needs to freeze the core of the atom in order to avoid variational instabilities. In other words, the Pauli approximation is not recommended for all electrons calculations of heavy atom. An alternative approximation of the Dirac equation is to consider that in the core region, we have E<< (2c2-V). At the zeroth order of approximation, the Hamiltonian takes the form:

).(22 2

2

2

20

!!!!!

"#$

+$

+= pVVc

cp

Vc

cpVH % (5)

This equation is known as ZORAxiii xiv(zero order regularization approximation). The ZORA approximation provides the possibility to perform all electrons calculation. ZORA can further be approximated by neglecting the last term (i.e. spin-orbit interaction) in equation (5). This is called scalar relativistic corrections, since only the spin-orbit term splits/couples non-relativistic representations. The scalar ZORA has been used in this work for geometry optimization. Nuclear shielding tensor. Using atomic units, the shielding tensorxv for a given nucleus, is defined as the second derivative of the total electronic energy, E, with respect to a constant external magnetic field, B, and a nuclear magnetic moment of nucleus with tth component µt

!

" st =#2E

#Bs#µtr B =

r µ =0

!

="

"Bs

#(r B ) ht

01+ Brhrt

11

r=1

3

$ #(r B )

r B =0

(6)

Represent the ground state electronic eigenfunction under the influence of the external magnetic

field. s,t refer to the tensor component According to ref. [7c], expression (6) can be split into paramagnetic, diamagnetic and spin orbit contribution.

so

st

d

st

p

stst !!!! ++= (7)

where

!!"

#$$%

&'++= (((

= )*+

+,

+µµ

,µ,µ,,µµ ---./

,

**

,

01

,

11

,

11

,2

1

2

1ii

N

ststst

occ

i

i

d

stddRhh

(8)

is the diamagnetic contribution of the shielding tensor,

( )

(9)Re2

)(2/

,

*

,

01,1

,

*

,

01

,

,1

,

*

,

01

!!"

#

$$%

&''(

)**+

,+''(

)**+

,-

.''(

)**+

,--=

/ / ///

/////

==

=

00

occ

i

vir

a

ai

N

v

vt

s

aiiij

N

v

vt

occ

ji

s

jiiii

N

tv

occ

i

i

p

st

ddhudd

hSddhRRci

123

34

3µ

µ

µ

123

34

3µ

µ

µ

123

34

3µµ

4µ4µµ

556

55655567

is the paramagnetic term and

)10('||Re2'||

'|)()2/(|

, ,,'

'*,1

,,'

'*,1

,

, ,,'

'*

!"

#$%

&+

'!"

#$%

&'((=

) ) ) )) ))

) ) )

==

***

=

occ

i

vir

a

v

SO

taj

s

aii

NSO

tij

k

ji

occ

ji

i

occ

i

SO

t

s

v

N

iiiSO

st

hdduhddS

hRRrcidd

+µ ,-..µ

.+

.µ

+µ ,-..µµ

.+

.µ

µµµ

+µ ,-..

.+

.µ

././0././0

././01

is the spin orbit shielding contribution, which can be split into occ-occ and occ-vir contributions. That is into summations including both occupied and occupied or occupied and virtual orbitals respectively. In eqs. (8) to (10)

),(4 32

11

, vtsQstvQ

Q

st rrrrrc

Kh !"=

##

$µ (11)

s

v

Q

tQtQ

s

v

Q

st RRrrc

iKprprRRr

rc

iKR !"

#$%

&'((+(!"

#$%

&'(=

))))))))))

)(4

)()()(4 3232

01

, µ*µ*µ*

3232

01

2)()(

2 Q

tQtQ

Q

trc

Kprpr

rc

Kh

!!!!

"+"=

where: K= [1-V/2c2]-1 i! Is a scale energy factor

Qµ is the nuclear magnetic moment attached to nucleus Q, rQ is the distance between the NMR nucleus Q and the reference electron,

'!

" id are coefficients of the complex atomic wave function and γ refers to spin α or β , φv refers to real atomic basis functions,

s

jiS,1 is the first order overlap matrix,

iu

a

ais

ia

ai

t

aiMrccu !!

""!!

""##$#

#$$

^

)0()0(

)0(

,

)0(

)0()0(

.1

)(2

1

)(2

1

%%&

'()

*+,

-.%

%&//

0 (12)

where: )0(

i! is the energy of occupied orbitals without external magnetic field but including spin-orbit contributions.

is the position vector to the center of the atomic orbital !" . The vector , occ refers to the number of occupied molecular orbitals (index i and j), vir the number of virtual orbitals (index a), and N is the total number of orbitals, µ and ν are indexes for basis functions, γ is an indice for alpha and beta spin

Assuming an equal occupation for all n occupied molecular orbitals ( for spin-polarized

systems or for closed-shell systems),i

uaM !!^

represents the coupling term between

virtual and occupied orbital with the magnetic operator uM

^

. The spin-orbit contribution is dominated by the Fermi-contactxvi term and is expressed as:

iNua

occ

i

vir

a

s

ia

FC

ts

so

tsrS

c

gu !"!

#$$ )0(

3

4 ^),1(

=%=& '' (13)

where uS

^

is defined as Cartesian component of the electronic spin operator, g is the electronic Zeeman factor. This contribution has been foundxvii important for density functional calculation of NMR chemical shifts in tungsten and lead compounds. A detailed description of all terms involved in the calculation of the shielding tensor can be found ref.7a,b Chemical Shift The chemical shift is simply evaluated as

)()()( samplerefsample !!" #= (14) Computational details The geometry optimization has been carried out for an isolated cluster and, for some complexes, in solvent media, using the ADF 2000 and 2002 packagexviii. The ZORA TZP basis set with frozen core on Pt (4d), Br (3p), and Cl (2p) has been used. ZORA relativistic corrections have been applied, (scalar in geometry optimizations and with spin-orbit for the calculation of the hyperfine tensor). LDA exchange-correlation functional (Slater for exchangexix and Vosko-Wilk-Nusair for correlationxx) has been retained for the geometry optimization. This is justified by the fact that the geometries obtained are closer to experiment than the generalized gradient approximations (GGA) let obtainxxi. Indeed, one obtains trends for the properties calculated at the GGA level of approximation (or hybrid functionals) on such geometries which are similar to those calculated at self-consistent GGA geometries. The solvation model retained for the study of solvent effects is the "conductor like screening model" (COSMO) model. It is well known that dielectric screening energies of a given geometry scale as

)/()1( x+! "" where ! is the permittivity of the screening medium and x is a parameter in the range 0-2 (for water ! =78.8 and x=0) In reality, for a conductor !"# , and screening in conductor can be handled easily. The total screening energy is classically given by the following expression using image charge methodxxii

QDQE 2/1!=" (15)

where Dij=R/(R2-rirj)2, and if the solvent effect is taken into account, the total energy of the conductor like system is given as

qAqQBqQCQE 2/12/1 ++= (16) where Q is a vector containing n point charges Qi at position ri within a sphere of radius R. Qi is

supposed to be enveloped by an arbitrary surface S and the screening energy can be obtained by dividing Qi into m small segment surfaces s with constant charge density and corresponding point charge qi. B (m*n matrix) is the electrostatic interaction of unit charges at rj with unit charges on s. C is the classical coulomb operator between the charges Qi A (mxm) is the electrostatic interaction of two different charge qi surface

Therefore by minimizing the equation (16) we can obtain the effective screening charge

q* = -A-1BQ=DQ. (17) and by replacing (17) into (16), the total energy of the screened system becomes

QQDQBBACQE *2/1)(2/1*)( 1!"!!=

!# (18) This equation is similar for the total screening energy of a conductor (equation 15) During the computational process, we can choose to include D* in the Hamiltonian of the system as a perturbation (post SCF calculation) or during the SCF procedure (Variation method). Three different types of cavityxxiii have been used to represent sphere surface model of the solute molecule in the solvent. These surfaces are constructed with GEPOL93 algorithmxxiv. "Asurf" yields the solvent accessible surface (SAS), it consists of the path traced by the center of a spherical solvent molecule rolling on the van der Waals surface. "ESurf" gives the solvent excluding surface (SES), which consists of the path traced by the surface of a spherical solvent molecule rolling on the VDW surface. "Klamt" as well as Esurf excludes also the cusp regions of the overlapping van der Waals spheres, but differs in the formulation. Since we do not have experimental values to compare with, the influence of solvent effect has been computed only for [PtCl6]-2 and [PtBr6]-2 complexes, for which, because of their Oh symmetry, only bond lengths can be altered. No angular distortion, nor any differential (Br vs Cl) charge transfer can be expected for the inclusion of these complexes into the solvent model. The 195Pt chemical shift has been estimated using the NMR program implemented in ADF using the wave function calculated with the ZORA relativistic approximation and including spin-orbit term. We have distinguished calculations performed with frozen cores and all electron calculations on 195Pt Results and discussion 1. Structure of the 195PtClxBr6-x

2- complexes The following table, table I gives the geometries used to compute the chemical shift.

TABLE I. Geometries used to compute chemical shift: SRZ=Scalar Relativistic ZORA, TZP=Triple zeta + polarization basis set. System Geometry Method used Pt-Cl bond

length Pt-Br bond length

[PtCl6]2- Optimized experimentala

SRZ/LDA/TZP gas elec. diffract.

2.341 2.317-2.334

[PtCl5Br] 2- Optimized

SRZ/LDA/TZP

2.341

2.484

trans [PtCl4Br2] 2-

Optimized

SRZ/LDA/TZP

2.340

2.489

cis [PtCl4Br2]2-

Optimized experimentalb

SRZ/LDA/TZP

2.334 2.358

2.488 2.471

meridian [PtCl3Br3]2-

Optimized

SRZ/LDA/TZP

2.345

2.489

facial [PtCl3Br3]2-

Optimized

SRZ/LDA/TZP

2.345

2.486

trans [PtCl2Br4]2-

Optimized

SRZ/LDA/TZP

2.334

2.489

cis [PtCl2Br4]2-

Optimized

SRZ/LDA/TZP

2.344

2.489

[PtClBr5]2- Optimized

SRZ/LDA/TZP

2.344

2.489

[PtBr6]2- Optimized

SRZ/LDA/TZP

2.488

a) Ref (23) b) Ref (24) Since the NMR experiments are performed in solution, no accurate comparison between the experimental structure and theoretical ones can be made; however, the comparison with solid-state structure, when available, indicate a rather good agreement with the theoretical structure obtained at the LDA level of approximation. 2. Solvent effects on the structure Table 2 gathers the results of the solvation effects as calculated through the 3 models, and the two perturbational/SCF approaches.

TABLE II. Influence of the model to compute solvent effect on the structure of the Complex A – Averaged Pt-Br bond length (Å) in [PtBr6]-2 complexes, (gas phase length: Pt-Br: 2.487 Å)

Surface model

Post SCF method (bond length Å)

Partial Charge (Mulliken)

Solvation energy (eV)

Variational Method bond length(Å)



Asurf Pt

2.519

0.6362

-5.6 2.476 0.6653 -5.63

ESurf Pt

2.525 0.6313

-6.7 2.472 0.5545 -7.19

Klamt Pt

2.524 0.6333

-6.27 2.466 0.6056 -6.47

B- Average of Pt-Cl bond length (Å) [PtCl6]2- complexes, (gas phase length: Pt-Cl: 2.341 Å)

Surface model

Post SCF method (bond length Å)



Variational Method (bond lengthÅ)



Asurf Pt

2.342 0.6018

-5.83 2.330 0.5907

-5.85

ESurf Pt

2.340 0.6041

-7.39

2.326

0.4330

-7.67

Klamt Pt

2.340 0.6046 -6.9 2.320 0.5511

-7.0

The bond length reported for the solvated complexes is an average of the six lengths obtained through the optimization process: this takes its origin in the fact that the excluding or accessible surface used in the solvation models does not retain the symmetry of the system, allowing a slight distortion to occur. It is interesting to notice that the variational procedure leads in all models to shorter bond lengths, by 10 to 21pm, according to the model retained. The change in Mulliken charges, on the contrary, is rather small with the Asurf model, a little bit larger with the Esurf model, and roughly in between with the Klamt model (the platinum charge decreases through solvation by 0.05 for PtCl6

2-, and 0.03 for PtBr62-

). This is related to the solvation energies which are, for the Klamt model, between the Asurf model (smaller) and the Esurf model (larger). The perturbational approach, on the contrary, which does not permit a change in the ionicity of the Pt-X bonds, is less sensitive, and leads to distortions in the opposite direction, if any. One cannot exclude a sensitivity of the effect to the basis set, which has not been studied.

3. NMR chemical shifts of the 195PtClxBr6-x2- complexes

As can be seen on Table III, a very good agreement with experiment is obtained provided that the shielding tensor is computed with an all electron basis set; PW91xxv gradient corrections has been used for exchange and correlation. On the contrary, the calculations with frozen cores exhibit a systematic deviation, which was expected because of the participation of core levels to the shielding which cannot be taken into account. One can notice that the chemical shift is dominated by the paramagnetic contribution whereas the diamagnetic contribution is rather small (0.5%). The paramagnetic chemical shift is largely determined by the first order coupling magnetic coupling term between occupied and virtual orbitals. Table III indicates also that as the number of softer ligands (Br) increases, through substitution of the harder ones (Cl), the chemical shift becomes more and more negative. Table III. Different contributions (paramagnetic, diamagnetic, spin-orbit/Fermi contact (SO/FC) to the 195Pt NMR chemical shift (ZORA formalism with all-electrons TZP basis sets). Values are given in ppm. Values obtained with the frozen cores approximation are in parenthesis. diff

exp!! "=

cal

Compound δ param δ diamag δ SO/FC δ calculated δ exp Diff [PtCl6]-2 0

(0) 0 (0)

0 (0)

0 (0)

0 0 (0)

[PtCl5Br]-2

-166.7 (-122.7)

0.6 (0.7)

-110.7 (-80.3)

-276.8 (-202.4)

-282 5.2 (79.6)

trans-[PtCl4Br2]-2

-326.4 (-241.7)

1.2 (1.4)

-258 (-187.1)

-583.3 (-427.5)

-583.6 0.3 (156.1)

cis-[PtCl4Br2]-2

-344. (-253.7)

1.2 (1.4)

-222.8 (-162.2)

-565.6 (-414.5)

-58.4 16.8 (167.9)

facial [PtCl3Br3]-2

-531.0 (-392.2)

1.8 (2.1)

-336.6 (-245.8)

-865.8 (-636.0)

-889.2 23.4 (253.2)

meridian [PtCl3Br3]-2

-516.5 (-383.0)

1.8 (2.1)

-372.6 (-271.2)

-887.6 (-652.1)

-891.4 3.8 (239.2)

trans [PtCl2Br4]-2

-699.5 (-520.6)

2.4 (2.8)

-526.1 (-383.3)

-1223.3 (-901.2)

-1210 -13.3 (308.8)

cis [PtCl2Br4]-2

-714.3 (-530.1)

2.4 (2.8)

-489.3 (-357.5)

-1201.2 (-884.9)

-1210 8.8 (325.1)

[PtClBr5]-2

-908.3 (-677.2)

3.0 (3.5)

-645.6 (-472.5)

-1551.7 (-1146.2)

-1540

-11.5 (393.816)

[PtBr6]-2

-1113.5 (-833.5)

3.6 (4.2)

-805.4 (-590.9)

-1915.2 (-1420.2)

-1870 -45.2 (449.8)

The necessity to perform all electrons calculations is clearly visible, and, as already said, was expected:

[195Pt-Cl_(6-x)-Br_x]2-

-2000

-1500

-1000

-500

0

0 1 2 3 4 5 6

number x of Br

ch

em

ica

l s

hif

t (p

pm

)

calc.

exp.

Diff

frozen core #

The chemical shift is roughly proportional to the number of bromine atoms, as can be seen from an inspection of Fig. 1 and from Table III. The assignment of the chemical shift to each isomer is experimentally straightforward because it can be extracted from the relative intensity of the peaks which is given by the statistical distribution of the isomers (e.g. PtX4Y2 exists with a theoretical ratio cis/trans = 4). Figure 1 shows an excellent correlation between experimental values and the computed values, the absolute maximum deviation being less than 2.3%, obtained for [PtBr6]2-, with an absolute average deviation being less than 1.5%. In order to check what could be the origin of this (small) deviation, we have calculated the influence of the bond length to the chemical shift. This is easy to perform on the [PtBr6]2- complex because, thanks to its Oh symmetry, there is only one bond length parameter. Figure 2 shows that one gets a quasi-linear relationship between the chemical shift and the bond length, with a slope of ca. 150 ppm/pm. One can notice (Table III) that the experimental difference in the chemical shift of the different isomers is rather small (1-2 ppm), and found somewhat larger theoretically (10-20 ppm). However the relative order between 2 isomers is always in agreement with the experiment. Finally, one have to recall that the chemical shift depends strongly to the geometry accuracy, the 1ppm difference in two chemical shifts corresponding to a difference of 0.01pm, i.e. much less than the precision required for our geometry optimization computation.

Chemical shift variation with bond length

312.8

163.4

12.4

-140.4

-200

-100

0

100

200

300

400

2.46 2.47 2.48 2.49 2.5 2.51

Pt-Br bond length

che

mic

al s

hift

Fig. 2. Variation of the chemical shift (ppm) with the Pt-Br bond length (Å) in [PtBr6]2-

Table IV. Contributions to the paramagnetic isotropic and the spin-orbit (Fermi contact) isotropic NMR shielding.

Compound Paramagnetic Isotropic NMR Shielding contribution (ppm)

Fermi contact Isotropic NMR Shielding contribution (ppm)

Valence- Virtual

Valence-Valence

total Valence- Virtual

Valence-Valence

total

[PtCl6]-2

0 0 0 0 0 0

[PtCl5Br]-2

-112.1 -54 -166.7 -76.7 -33.9 -110.7

trans-[PtCl4Br2]-2

-216.8 -109.8 -326.4 -181.3 -76.8 -258

cis-[PtCl4Br2]-2

-237.8 -106.5 344 -156.1 -66.7 -222.8

facial [PtCl3Br3]-2 C3V

-377.1 -154 -531 -263.8 -108.8 -372.6

meridian [PtCl3Br3]-2

-358 -159.1 -516.7 -236.8 -108.8 -372.6

trans [PtCl2Br4]-2

-490.4 -209.7 -699.5 -375.9 -150.3 -526.1

cis [PtCl2Br4]-2

-511.2 -203.6 -714.3 -350.8 -138.5 -489.3

[PtClBr5]-2

-659 -250 -908.3 -467.5 -178.1 -645.6

[PtBr6]-2

-821.5 -292.8 -1113.5

-589.3 -216.1 -915.2

As indicated in equations 9-10, it is interesting to compare the importance of the contributions to the isotropic NMR shielding, coming from orbitals excitations either in the paramagnetic isotropic term or the spin-orbit (Fermi contact) term. One sees in Table IV that roughly 2/3 of the effect originates from the paramagnetic contribution. Whereas the diamagnetic contribution to the screening tensor is rather modest. It does not involve the virtual orbitals (eq. 8) and hence is not sensitive to the gap and remains unchanged among the different clusters. Thus, it does not contribute significantly to the chemical shift. 4. Solvent effects on the NMR chemical shift Because the experiments are performed in aqueous solution, one can expect that it is necessary to take solvent effects into account. As already said, we have restricted this study to the [PtCl6]-2 (references) and to the [PtBr6]-2 cases, conserving the octahedral symmetry of the complexes. Other molecules

would exhibit distortions which may involve different polarizabilities of the Pt-Br and Pt-Cl bonds, and therefore possibly be more sensitive to other parameters such as basis sets. We have reported in Table V the results obtained for the different cavity models, and the perturbational/SCF calculations. Table V: deviations of the chemical shift (ppm) for each type of cavity. Surface option Pertubation approach Variational approach Asurf 409 -27 ESurf 804 -53 Klamt 523 4 The results show unambiguously that it is necessary to use a variational process. The perturbation approach leading in some cases to deviations with a wrong sign. Finally no test has been performed in order to check is the triangulation of the van der Waals spheres uses a sufficient number of sections. It is fixed to 5 in the ADF code, and is assumed to be accurate enough by the developers. From this table, it appears that the Klamt model provides the best results, as expected on the basis of the general trends derived from other worksxxvi,xxvii,xxviii 5. Influence of the basis sets and the exchange-correlation functionals Both of them have been tested. The results obtained indicate that one obtains a very accurate determination of the chemical shift, with an average deviation as small as 0.1 % (average absolute deviation less than 1.5%) when a TZP basis set is chosen. This is clearly visible in the Figure 1. Calculations performed with TZ2P basis sets did not lead to any significant improvement brought by the addition of a second polarization function into the basis set. Indeed slightly worsen results, with an average absolute deviation amounting to 3 % have been obtained. This is not too surprising, since the environment of the platinum atom is highly symmetric in the whole set of complexes studied. The results obtained with various GGA exchange-correlation functionals belonging to the GGA family are reported in Table VI.

Table VI: Algebraic deviation (with respect to experiment) of the chemical shift for several GGA exchange-correlation functionals.

GGA System

PW91 PBExxix BLYPxxx,xxxi OLYPxxxii,30 OPBE28,31 revPBExxxiii RPBExxxiv

[PtCl6]2- 0 0 0 0 0 0 0 [PtCl5Br]2- 5.2 5.6 -2.8 -7.1 5. 3.1 1.3 trans [PtCl4Br2]2-

0.3 2.1 -17.5 -22.2 4.1 -2.3 -5.3

cis [PtCl4Br2]2- 16.8 17.3 1.7 -6.7 17.1 12.4 8.9 meridian [PtCl3Br3]-2

23.4 5.4 -21.3 -28.8 8.9 -1 -5.9

facial [PtCl3Br3]-2

3.8 23.7 0.2 -10.5 24.7 16.6 11.2

trans [PtCl2Br4]-2

-13.3 -11.1 -47.9 -54.2 -2.8 -18.8 -25

cis [PtCl2Br4]-2 8.8 9.7 -23.5 -33.1 15.6 1.5 -5.2 [PtClBr5]-2 -11.7 -0.6 -52.3 -59.8 2.3 -18.7 -26.7 [PtBr6]-2 -45.2 -43.8 -95.6 -100.2 -25.0 -53.6 -63.0

The results show a limited influence on the exchange-correlation functional employed. However, the functionals derived from a combination with the given correlation functional (namely PBE or LYP) internally consistent i.e. very similar among them. This would indicate that the dynamic correlation, modelled through the correlation functional plays an important role, whereas the exchange functional, which describes the non-dynamical correlation, is less important. Concluding remarks The uncoupled GIAO-DFT method with ZORA relativistic corrections have lead to an excellent description of the 195Pt NMR spectra of 195PtClxBr6-x

2- complexes. The frozen core approximation leads to a deviation of approximately 20%, whereas the full electron calculation provides an agreement with experiment which is extremely good, and sensitive to accurate determinations of the bond lengths of the complexes. These bond lengths are slightly modified by solvation, as expected, and the Klamt model is the one which leads to the best agreement with experiment. The present calculations have been obtained using ADF like black box. That is the almost linear dependence on the number of substituted bromide ions is not yet understood. We plan to address this problem deeper in a forthcoming publication.

References i A. von Zelewsky, Helv. Chim. Acta. 1968, 803- 807 ii, R.R. Dean, J.C. Green, J. Chem. Soc. A 1968, 3047-3050 iii N. F. Ramsey, Phys. Rev. 1950, 78, 699-703 iv M. Kaupp, V.G. Malkin, O.L. Malkina, D. Salahub, J. Am. Chem. Soc. 1995,117, 1851 -1852 v V.G. Malkin, O.L. Malkina, L.A. Erikson, D.R. Salahub,

In Modern Density Functional Theory: A Tool for Chemistry; Politzer, P.,

Seminario, J. M., Eds.; Elsevier: Amsterdam, the Netherlands, 1995; Vol.2. vi W. Kutzelnigg, U. Fleisher, M. Schindler, In “NMR-Basic Principles and Progress”; Springer-

Verlag: Berlin, 1990; 23, 165. vii a)H.G. Schreckenbach, T. Ziegler, J. Phys. Chem. 1995,99, 606-611

b) H. G. Schreckenbach, Relativity and Magnetic Properties.A Density Functional Study Ph.D. Thesis

(University of Calgary, Aberta, Canada, September 1996)

c) S. K. Wolff, T. Ziegler, E. van Lenthe, E. J. Baerends, J. Chem. Phys.1999, 110, 7689-7698 viii T.M. Gilbert, T. Ziegler, J. Phys. Chem. A 1999, 103, 7535-7543 ix C.J. Pickard, F. Mauri, Phys. Rev B. 2001, 63, 245101-245112 x P.E. Blöchl, Phys. Rev .B 50, 1994, 17953–17979 xi A. Hesse, H. Meir, B. Zeeh, Spektroscopische Methoden in Organische Chemie, G. Thieme Verlag Stuggart, 1991,

4.auflage xii A. Streitwieser Jr. U. Müller-Westerhoff, J. Am. Chem. Soc. 1968, 90 7364-7364 xiii E. van Lenthe, E.J. Baerends, J. Snijders J. Chem. Phys. 1993,99 ,4597- 4610 xiv E. van Lenthe, E.J. Baerends, J. Snijders J. Chem. Phys. 1996,105 ,6505-6516 xv H. Fukui, Magn. Reson. Rev.1987, 11, 205. xvi S.K. Wolff, T. Ziegler, J. Chem. Phys. 1998, 103, 895-905. xvii A. Rodriguez-Fortea, P. Alemany, T. Ziegler J. Phys. Chem. A. 1999, 103, 8288-8294 xviii SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands,

http://www.scm.com

E.J. Baerends, J. Autschbach, A. Bérces, C. Bo, P. M. Boerrigter, L. Cavallo, D.P. Chong, L. Deng, R.

M. Dickson, D. E. Ellis, L. Fan, T. H. Fischer, C. Fonseca Guerra, S. J. A. van Gisbergen, J. A.

Groeneveld, O. V. Gritsenko, M. Grüning, F. E. Harris, P. van den Hoek, H. Jacobsen, G. van Kessel,

F. Kootstra, E. van Lenthe, D.A. McCormack, V. P. Osinga, S. Patchkovskii, P. H. T. Philipsen, D.

Post, C. C. Pye, W. Ravenek, P. Ros, P. R. T. Schipper, G. Schreckenbach, J. G. Snijders, M. Sola, M.

Swart, D. Swerhone, G. te Velde, P. Vernooijs, L. Versluis, O. Visser, E. van Wezenbeek, G.

Wiesenekker, S. K. Wolff, T. K. Woo, T. Ziegler xix J.C Slater, Phys. Rev. 1951, 81 385-390. xx S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 1980,58, 1200-1211 xxi H. Chermette, Coord. Chem. Rev., 1998, 699.178-180, xxii J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. xxiii A. Klamt, G. Schüürmann, J. Chem. Soc. Perkin Trans 2. 1993, 5, 799-805 xxiv J. Pascual-Ahuir, J.L. E.Silla, I. Tuñon, J. Comput. Chem. 1994, 15, 1127-1139 xxv J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson , C. Fiolhais, Phys. Rev. B.

1992,46, 6671-6687 xxvi A. Klamt, J. Chem. 1995, 99, 2224-2235 xxvii J. Andzelm, C. Kölmel, A. Klamt: , J. Chem. Phys. 1995, 103, 9312-9320 xxviii K. Baldridge, A. Klamt:, J. Chem. Phys. 1997, 106, 6622-6633 xxix J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 1996, 77 3865-3868 xxx A.D. Becke, Phys. Rev A. 1988 38 3098-3100 xxxi C. Lee, W. Yang, R.G. Parr, Phys. Rev. B. 1988, 37 , 785-789 xxxii N.C. Handy, A.J. Cohen Mol. Phys. 2001, 99 403-412, xxxiii Y. Zhang, W. Yang, Phys. Rev. Lett. 1998 ,80 890-890 xxxiv B. Hammer, L.B. Hansen, J.K. Norskov, Phys. Rev. B. 1999 59 7413-7421

Lecture 5: First principle calculation of the NMR spin...

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