9.1Introduction

Various methods for the calculation of the nuclear magnetic shielding have beendeveloped and applied since the fundamental formulation of the theory by Ramsey[1]. Originally, most of the methods for “ab initio” or “first principle” treatments ofNMR parameters in molecules were based on the Hartree–Fock (HF) formalism inconnection with perturbation theory to consider the external magnetic field. Linearcombination of atomic orbitals (LCAO) “ab initio” calculations with large basis setshave been performed within the coupled HF perturbation theory (CHF) (see e.g.Ref. [2]). The effect of electron correlation was studied and NMR schemes were de-veloped for many post-HFmethods (for review, see Ref. [3]).

Significant improvements for the calculation of the NMR parameters could beachieved by the GIAO (gauge including atomic orbitals, originally called “gaugeinvariant atomic orbitals” or “London orbitals”) approach from Ditchfield [4], as wellas by the method of individual gauge for localised orbitals (IGLO) from Kutzelniggand Schindler [5]. Since the early 90s, both methods (see, for example, for IGLO[6, 7], for GIAO [8, 9]) have been also successfully applied within density-functionaltheory (DFT) and the local density approximation (LDA) or the generalised gradientapproximation (GGA) – for reviews see e.g. Refs. [10, 11]. However, it has beenpointed out already by Bieger et al. [12] that the calculation of the nuclear magneticshielding cannot strictly be justified within DFT, as the presence of the magneticfield requires an extension. This extension is given by the CDFT (current densityfunctional theory) [13].

The application of semiempirical quantum chemical methods for NMR calcula-tions goes back to the 1960s. After studies with simple model wavefunctions, alreadyback in the 1950s, the semiempirical NDO (neglect of differential overlap) methods[14] were used in connection with an uncoupled Hartree-Fock perturbation treat-ment (UCHF) [15]. This is also known as the “sum-over-states” (SOS) method [16].Combinations of the complete NDO (CNDO) and intermediate NDO (INDO) meth-od with GIAOs were also developed and applied [17, 18], as well as combinationswith the method of individual gauge transformations [19, 20].

141

9

Semiempirical Methods for the Calculation of NMR ChemicalShifts

Thomas Heine and Gotthard Seifert

Calculation of NMR and EPR Parameters. Theory and Applications.Edited by Martin Kaupp, Michael B�hl, Vladimir G. MalkinCopyright � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-30779-6

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts

Most widely used was a rather crude model for estimations of nuclear magneticshieldings, the so called “DE” model. This model can formally be derived from theUCHF treatment, where the energy denominator in the perturbation expansionwith the virtual orbitals is just replaced by an “averaged” energy denominator (DE).Though it is a very strong approximation, it was successfully applied for qualitativeinterpretations of chemical shifts, see e.g. Ref. [21]. It can give simple relations be-tween chemical shifts and corresponding charges of atoms obtained from quantumchemical calculations, but it also makes clear that there are generally no simple cor-relations between chemical shifts and charges, for example.

After a short description of the basics for calculating nuclear magnetic shieldingin (closed shell) molecules, the “sum-over-states” (SOS) as well as the “DE” modelare described briefly. As examples of modern semiempirical schemes the GIAO-MNDO (modified NDO) [18, 22] and the IGLO-DFTB [23] methods are discussed.The capabilities of these methods are illustrated by a few representative applications.Finally, concluding remarks concerning the outlook and limitations of semiempiri-cal calculations of nuclear magnetic shieldings are given. We will not discuss herethe manifold of models, which are not based on quantum chemical calculations.

9.2Methods

9.2.1General Considerations

The nuclear magnetic shielding tensor (rk « rabk ) describes the induced field (~BBind)

at a nucleus k in presence of an external field (~BB0 ):

ð~BBindÞa ¼ �Xb

rkabð~BB0Þb ð9:1Þ

Biot-Savart’s law gives the relation between the induced field and the current densityin a molecule (Gaussian units):

~BBind ¼ 1c

Z ~JJ �~rrkr3k

d3r ð9:2Þ

The current density (~JJ) is given by the electronic wavefunction:

~JJ ¼ i2

ðrW� ÞW�W

�ðrWÞ� �

� 1c~AAW

�W: ð9:3Þ

~AA is the vector potential of the external magnetic field. According to the definition ofthe shielding tensor as a linear response property, only terms in first order of ~BB0 areneeded in a Taylor series of J:

142

9.2 Methods

Jð~BB0Þ ¼~JJ0 þ~BB0

@~JJ

@~BB0

þ ::: (9.4)

~JJ0, the current density in the absence of an external magnetic field, is zero for mole-

cules with no permanent magnetic moment. The first order current density is thengiven by:

~BB0 J1 ¼ i

2ðrW

0ÞW1 �W1ðrW

0Þh i

� 1c~AAW

0W

0: (9.5)

I.e., the first order current density ~BB0 J1 � @~JJ

@~BB0

� �with the first order perturbed

wave function (W1), has to be considered in Biot-Savart’s law for calculations ofnuclear magnetic shielding tensors. Within a Coulombic gauge for the vector poten-tial (~AA ¼ 1=2ð~BB�~rrÞ), one obtains finally for the shielding tensor at nucleus k:

r ¼ 12c

2 W0���~rr~rrk I �~rrk �~rr

r3k

���W0

* +� 2

cW

0��� ~LLr3k

���~WW1

* +ð9:6Þ

The first term is the so-called diamagnetic contribution (rdia) of the shielding tensor,whereas the second term is called the paramagnetic term (rpara). Expanding the per-turbed wavefunction (W1) in terms of the excited states of the unperturbed wave-function (W

0

n) one obtains:

rpara ¼ � 1

c

Xn 6¼0

1En � E0

W0n

��� ~LLr3k

���W00

* +W

00

���~LL���W0n

D E; ð9:7Þ

where E0 and En are the ground state and the excited state energies, respectively, and~LL is the angular momentum operator. The diamagnetic term remains unchangedbecause it depends only on the unperturbed wavefunction. This is the “classical”representation of the nuclear magnetic shielding tensor in diamagnetic moleculesas already derived by Ramsey [1]. It is also the basis of most of the semiempiricaltreatments of (nuclear) magnetic shieldings in molecules.

9.2.2The SOS and the “DE” Model

Using a simple molecular orbital (MO – wi) expansion for the wave functions (Wn),and an LCAO representation of the MOs (wi ¼

Plcilul), the nuclear magnetic

shielding tensor may be written as:

r ¼ 12c

2

Xocci

Xlm

cilc

im ul

���~rr~rrkI �~rrk �~rr

r3k

���um

* +�

143

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts

� 2c2

Xocci

Xunoccl

ðei � el Þ�1X

l<m

Xk<g

cllc

im � c

ilc

lm

� �clkc

ig � c

ikc

lg

� ��

� ul

���� ~LLr3k

����um

* +uk

���~LL���ug

D Eð9:8Þ

The summation includes occupied (“occ”) and unoccupied (“unocc”) MOs and isoften called the “sum-over-states” (SOS) method [16]. A very crude estimation of thenuclear magnetic shielding tensor is possible, if the summation over the occupiedand unoccupied MOs in the paramagnetic term is replaced by an averaged energydenominator (DE) [24]:

rpara ¼ � 2

c2

1DE

Xocci

Xlm

cilc

im ul

���~LL �~LLr3k

���um

* +ð9:9Þ

The diamagnetic term remains again unchanged. Often, DE is correlated with theHOMO–LUMO gap, or experimental excitation energies, or just taken as a fixedempirical parameter. In this approximation, the paramagnetic term also containsonly contributions from occupied MOs.

9.2.3The CHF Equations within Semiempirical Schemes, GIAO-MNDO

After some early trials [17] and several attempts in between [25–28] to apply theGIAO treatment within the semiempirical NDO methods, recently Patchkovskii andThiel [18, 22] proposed a combination of the MNDO method with GIAOs. In thismethod, GIAOs [4] have been introduced to ensure gauge invariance

jl ~rrð Þ ¼ ul ~rrð Þ exp � i2c

~BB�~RRl

� �y~rr

� �; ð9:10Þ

where ul is a real atomic basis function (e.g., Slater function or Gaussian), centeredat a position ~RRl. The shielding is expressed as the second derivative of the energywith respect to the magnetic field and the magnetic moment of the nucleus, whichleads to the CHF equations within the MNDO approximation (for details, see Ref.[22]). The wavefunction perturbed in ~BB is expressed using first-order perturbationtheory in an expansion of unoccupied states. The phase factors of the GIAOs cancelfor all one- and two-center two-electron integrals within the MNDO approximation.The magnetic field enters the one-electron Fock matrix contribution in the CHFequations, which has the form

144

9.2.3 The CHF Equations within Semiempirical Schemes, GIAO-MNDO

Fa;alm ¼ H

a0lm ¼ @

@~BBa

jl

���hh00 þ~BBahha0���jm

D E���~BB¼0

¼ ul

���hha0 ���um

D E� i2c

~rr �~RRl

� �aul

���hh00 ���um

�

þ i2c

ul

���hh00 ��� ~rr �~RRl

� �aum

� ð9:11Þ

which is notably simpler than in ab initio theory. NDO-type approximations furthersimplify this expression to

Ha0lm ¼ 1

2c

(~RRl �~RRm

� �H

MNDOlm

þ ~RRl �~RRm

� �� b

Nlm ul

���~rr �~RRm

���um

D E� �

� 12

ul

���LLRm���um

D Eþ 12

um

���LLRl���ul

D E)a

: ð9:12Þ

The equivalent expression (except for having bNlm ¼ blm as a parameter which can bevaried independently) was used previously in INDO [25] and MNDO [26] studies ofthe shielding tensors. An earlier INDO study [27] used a similar expression, butomitted the second term, while an uncoupled MNDO study [28] retained only thetwo last terms.

Finally, the GIAO-MNDO equations can be written as (gauge-independent) dia-and paramagnetic contributions:

rdiaab ¼

Xlm

HabðdÞlm Plm ð9:13Þ

rparaab ¼

Xlm

HabðpÞlm Plm þ

Xlm

H0blmP

a0lm ; ð9:14Þ

H0blm ¼ � 1

cul

����� LLRb��~rr �~RR��3�����um

* +; ð9:15Þ

HabðpÞlm ¼ � 1

2c~RRl �~RRm

� �aH

0blm ; ð9:16Þ

HabðdÞlm ¼ 1

2c2

(ul

������

~rr �~RRl

� �� ~RRm �~RRl

� ��a

LLRb��~rr �~RR��3�����um

* +

þ ul

����� dabð~rr �~RRmÞ

yð~rr �~RRÞ � ð~rr �~RRmÞbð~rr �~RRÞa��~rr �~RR��3

�����um

* +); ð9:17Þ

145

where ~RR is the position of the nucleus (or point in space) at which the shieldingtensor is evaluated. These equations completely define the NMR shielding tensorwithin the MNDO approximation.

GIAO-MNDO parameters have been created for H, C, N, and O, and tested for anextensive set of molecules [18, 22]. Linear correlation with experimental data gavecorrelation coefficients of 0.994 for 13C NMR parameters. In the parametrisation ofPatchkovskii and Thiel, which uses nine parameters for four elements, the RMSvalue for 13C NMR chemical shift is 12.6 ppm (out of a test set of 94 molecules) [22],which is considerably lower than in earlier GIAO-MNDO treatments. However, indi-vidual differences of 50 ppm have been observed for some sp3 carbons, and theseerrors can become larger than 100 ppm for shifts in the high upfield region. Theperformance of 17O and 15N shifts is similar, but for 1H shifts the results can beimproved to an individual error of ~3 ppm if three-center integrals are taken intoaccount in the MNDO treatment [22].

9.2.4IGLO-DFTB, a Semiempirical Method within the CDFT

The density-functional based non-orthogonal tight-binding (DFTB) method [29, 30]is an approximation to DFT. The DFTB method is based on an LCAO Ansatz for theKohn–Sham molecular orbitals. The expansion coefficients cim are found by solvingthe secular problem

Xl

cil Hlm � eiSlm �

¼ 0 for all m ð9:18Þ

which is expressed in terms of the Kohn–Sham matrix elements Hlm=Æul|TT+Veff|umæand overlap matrix elements Slm=Æul|umæ. The effective potential Veff is approximatedas a superposition of atomic contributions, each determined by an LDA–DFT calcu-lation on a fictitious spherical pseudo-atom subjected to an additional potential(r/r0)

n. The valence wavefunctions and the effective potential are both taken fromthe pseudo-atomic calculation. It is only necessary to consider two-center elementsof the Kohn–Sham matrix [29]

Hlm ¼hul jTT þ Vj þ Vk jumi for j 6¼ kel for l ¼ mo otherwise;

8<: (9.19)

containing the effective potentials Vj, Vk of the atoms j and k that carry functions ul

and um. In the case of j = k, the one-particle energies of the free atom el are used,giving the correct reference energy in the dissociation limit. Restriction to two-cen-ter terms leads to a Kohn–Sham matrix similar to empirically parametrised non-or-thogonal tight-binding schemes, but all parameters are obtained here consistentlyfrom LDA–DFT calculations.

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts146

In the present IGLO-DFTB implementation, the atomic orbitals and the atomicelectron density are expanded in a set of contracted Gaussian-type orbitals (CGTO),which are computed in a modified version of the program package deMon [31], butthis way the GTO-DFTB method can be implemented in any conventional wave-function-based package for property calculations in quantum chemistry.

So far, it has been applied with the deMon-NMR package [6,7], which uses theIGLO method [5]. This choice has the advantage of reducing the gauge dependenceand the paramagnetic contributions, thereby minimising the sensitivity to basis setsin the final computed result. As in present DFT-NMR treatments, IGLO-DFTB is anuncoupled theory. For the computation of the shielding tensor the deMon-NMRpackage [6] is used. Localised molecular orbitals were constructed with the iterativeFoster–Boys procedure [32] which gives a relatively strong localisation. In the IGLO-DFTB method, only valence orbitals are treated. Fullerenes with their half-fullvalence shells are favourable cases as their virtual spaces usually span all necessarysymmetries of the perturbed molecular orbitals [33].

Chemical shifts are calculated from the calculated shieldings of the fullerene siteand C60 and then transformed to dTMS values. The same technique is applied in abinitio computations of chemical shifts of fullerenes to remove a systematic error(see Chapter 25 by Heine).

The raw calculated 13C NMR spectra span too large a range of shifts in compari-son to experiment. This feature arises from an overestimation of the spread of theparamagnetic parts of the shieldings. We therefore apply an empirical correction,multiplying the paramagnetic part of all shieldings by a constant scaling factor. Thisis equivalent to making a correction of the energies of the unoccupied molecularorbitals. It is well known that approximate DFT treatments, including many LDAand GGA functionals, give poor energies for unoccupied orbitals [34] and more-or-less sophisticated correction schemes have been proposed [7]. The simple multipli-cative factor chosen here is 0.7, as this gives a reasonable value for the sp2–sp3 differ-ence in the C60 dimer.

9.3Representative Applications

A major field of recent application of semi-empirical methods for the evaluation ofNMR parameters have been fullerenes and their derivatives (see also Chapter 25).These are relatively large molecules (60 and more atoms, hence 240 and more basisfunctions in a minimal basis treatment), and usually several isomers needed to betreated. Two questions have been addressed in the literature using these methods:Nucleus-independent chemical shifts [35] of the centers of organic rings [36] andfullerenes [37], and the computation of 13C NMR chemical shifts of fullerenes andtheir derivatives [23, 38–40].

9.3 Representative Applications 147

9.3.1NICS of Organic Molecules

GIAO-MNDO was used to compute the NICS for a wide range of organic molecules,including [n]-annulenes, polycyclic hydrocarbons, heterocycles, cage molecules, ful-lerenes, and pericyclic transition states [36]. In general, there was reasonable agree-ment with NICS data from ab initio and density functional calculations, which isvisualised in Fig. 9.1.

The semiempirical NICS values tend to be smaller in absolute value than their abinitio counterparts, but they often show similar trends. The aromatic or antiaromaticcharacter of a given system can normally be assigned correctly on the basis of theMNDO NICS values.

9.3.2Endohedral NICS Values in Fullerenes

NICS values in the centers of fullerenes have been discussed as a useful criterionfor the aromaticity of fullerenes [41]. The NICS values at cage centers are essentiallyequal to the experimentally accessible 3He endohedral shift, which is one possibilityfor the characterisation of fullerenes using a single index (for a review, see Ref. [42]).Chen and Thiel [37] evaluated endohedral NICS values for a large series of fuller-enes, including small fullerenes C20 to C50, and fullerenes with isolated pentagons

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts148

-80

-60

-40

-20

0

20

40

60

-80 -60 -40 -20 0 20 40 60

MN

DO

/MB

NIC

S,

pp

m

Ab initio NICS, ppm

ideal4-membered rings5-membered rings6-membered rings

7+-membered rings3D cages

above/below ringtransition structures

Figure 9.1 Correlation between MNDO and ab initio NICSvalues (data from Ref. [36]). This figure has been kindly providedby Dr. Serguei Patchkovskii.

C60 to C102. In these computations, the geometry of the cages has been computed byMNDO and B3LYP/6-31G* levels. The endohedral NICS values have been evaluatedusing the GIAO-MNDO and using GIAO-CHF/3-21G for the higher, and GIAO-CHF/6-31G* for the lower fullerenes. Thus, the influence of the computational levelof the geometry and of the shielding computation on the chemical shift can be eval-uated.

The results are compared in Fig. 9.2. It can clearly be seen that the correlationbetween MNDO and CHF shifts is much better for the higher than for the lowerfullerenes. Two possibilities can explain this result: (i) the MNDO method worksbetter for the higher fullerenes due to their larger HOMO–LUMO gap and higherstability, and/or (ii) this fact is due to the relatively small 3-21G basis set, which hasonly one s and one p function more than the minimal MNDO basis itself, whichwas used for the CHF computations for the higher fullerenes.

9.3.3C119, an Odd-Numbered Fullerene

Fullerenes contain even numbers of trivalent, sp2-like atoms [43]. However, McEl-vany et al. [44] observed additional odd-numbered species C119, C129 and C139 bymass spectroscopy, which are expected to be dimeric structures of C60 and C70 fuller-enes, where several sp3-like bridging atoms link the two moieties with overall loss ofone atom [45]. These molecules can be produced e.g. by reacting fullerenes withozone [44] or by thermolysis of C120O at 550–600 �C with a yield of ~1% [46]. TheC119 species has a single predominant isomer for which the 13C NMR spectrum wasrecorded by Gromov et al. [46]. The observed pattern is compatible with an isomer ofC2 symmetry with two signals in the sp3 carbon region accounting for three (2+1)atoms.

In agreement with other calculations, the DFTB method finds the isomer A (C2)to be favoured energetically by ~1.9 eV among eight plausible candidates for thestructure of C119 [39]. The calculated 13C NMR chemical shifts [39] are compatiblewith the experimental spectrum, and identify this isomer as the species obtained

9.3 Representative Applications 149

-80 -60 -40 -20 0 20

-80

-60

-40

-20

0

20

40

DF

T o

r M

ND

O N

ICS

(p

pm

)

ab initio NICS (ppm)

DFT//DFT

y=1.11x+14.18 R2=0.88 SD=6.82

MNDO//DFT

y=0.40x+3.51 R2=0.74 SD=4.24

MNDO//MNDO

y=0.35x+4.46 R2=0.66 SD=4.37

-35 -30 -25 -20 -15 -10 -5 0

-16

-14

-12

-10

-8

-6

-4

-2

0

2

MN

DO

NIC

S (

pp

m)

ab initio NICS (ppm)

MNDO//DFT

y=0.45x-0.19 R2=0.86 SD=0.96

MNDO//MNDO

y=0.34x-1.53 R2=0.82 SD=0.85

Figure 9.2 Scatter plot of semiempirical/DFT vs. ab initio NICSvalues (in ppm), (a) for higher, (b) for smaller fullerenes. Theseplots have been kindly provided by Dr. Z. Chen.

experimentally by Gromov et al. [46]. The structures and 13C NMR data are shownin Fig. 9.3.

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts150

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

1

170 150 130 110 90 70 500

2

170 150 130 110 90 70 500

1

Figure 9.3 Structures (top and side view) and 13C NMR pat-terns of proposed C119 isomers are displayed on the left, theidealised experimental [46] (top) and computed 13C NMR pat-terns on the right side. Displayed data is taken from Ref. [39].NMR chemical shifts are given in ppm with respect to TMS.

9.4Concluding Remarks: Limitations of Semi-Empirical Methods for the Calculationof NMR Parameters

In all present implementations NMR parameters are computed as a sum represent-ed by a diamagnetic and a paramagnetic part. The diamagnetic part is simply anexpectation value, while the paramagnetic part is calculated over an expansion seriesof unoccupied molecular orbitals. While the diamagnetic part can be calculated atsatisfactory accuracy in semi-empirical methods, the paramagnetic part suffers thelimitations of the expansion series by the fact that a minimal valence basis is used.In semi-empirical methods it is not straightforward to go towards a basis set limit,as the change of basis will influence the parametrisation of the method itself. Inaddition to the well-known removal of gauge dependence it is therefore absolutelynecessary to reduce the paramagnetic contribution by an efficient gauge transforma-tion, as for example IGLO or GIAO.

For the same reasons, those applications work best for semiempirical methodswhich have either a small paramagnetic contribution (for example which are nearlyspherical symmetric, such as noble gases), or large systems with a large number ofvirtual basis functions, which may compensate for the drawback of the minimalbasis to some extent. Similar comments can be made for ab initio methods employ-ing small basis sets, and the quality of semi-empirical results for fullerenes are com-petitive with CHF-computations at the 3-21G level (see the fullerene dimer discus-sion in Chapter 25.).

With the success of density-functional theory, the availability of inexpensive andperformant computational hardware and the development of highly efficient algo-rithms for quantum-chemistry software, the importance of semi-empirical methodswithin quantum chemistry has been strongly reduced. Still, these types of methodshave their niche, for example for orienting computations, evaluation of trends, com-parison of many isomers and computation of very large molecules.

This conclusion may have been given for any other type of structure, energy orproperty computation with semi-empirical methods. However, generally for NMRcomputations only one single point calculation is necessary, and in most cases thisis a feasible task, even though it may take a long computing time.

Acknowledgements

We thank Dr. Serguei Patchkovskii for many fruitful discussions. The Deutsche For-schungsgemeinschaft is acknowledged for financial support.

9.4 Concluding Remarks: Limitations of Semi-Empirical Methods 151

9 Semiempirical Methods for the Calculation of NMR Chemical Shifts152

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