Density functional calculation of nuclear magnetic resonance chemical shifts

Transcription

1 Density functional calculation of nuclear magnetic resonance chemical shifts Christoph van Wüllen Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D Bochum, Germany Received 22 August 1994; accepted 24 October 1994 A current-density functional theory for the calculation of nuclear magnetic resonance chemical shifts is presented. If the Kohn Sham orbitals are expanded in a finite basis set, one of the main problems is the strong dependency of the results with respect to a shift of the gauge origin of the vector potential which describes the external magnetic field. Two computational schemes implementing both the individual gauge for localized orbitals IGLO and gauge including atomic orbitals GIAO concepts, which overcome this problem by introducing distributed gauge origins, are presented in detail. A comparison of the density functional IGLO and GIAO schemes shows that IGLO is much more efficient if one neglects the current-dependent part of the density functional as is done in uncoupled density functional theory, but that this advantage is less pronounced in the full current-density functional treatment American Institute of Physics. I. INTRODUCTION The calculation of nuclear magnetic resonance NMR chemical shifts using density functional theory DFT does not have a long history. It started in 1985 within the X approximation. 1,2 If an exchange-correlation functional depending only on the density is used, the resulting equations resemble those of uncoupled Hartree Fock and the method might be called uncoupled DFT. Later on, attempts were made to overcome basis set insaturation problems using the gauge including atomic orbitals GIAO Ref. 3 and individual gauge for localized orbitals IGLO Ref. 4 concepts. However, as is now well known, 5,6 the Hohenberg Kohn theorem no longer holds in the presence of a magnetic field and an extension of DFT to a current-density functional theory CDFT is necessary. The exchange-correlation energy then is a functional of the density and the paramagnetic current density j p, which are, for a closed-shell Slater determinant with doubly occupied orbitals i atomic units and Einstein summation convention are used throughout this paper, 2 i * i, j p 2 i 2 i * i i i *. Vingale and Rasolt 6 have shown that there is no gaugeinvariant theory with a density functional depending locally on and j p since j p is not a gauge-invariant quantity. It can be expressed through the gauge-independent total current density j and the external vector potential A, j p j 1 c A. A gauge transformation A A with any scalar function changes j p by j p j p 1 c Noting that 0, Vingale and Rasolt 6 introduced a new, gauge-invariant variable, j p 4 and showed that any exchange-correlation functional giving a gauge invariant theory can be expressed as a functional of and. They also have introduced a current-density exchange-correlation functional within the local density approximation, which takes the form E xc,e xc,0 g 2 dr. As mentioned, it is not possible to find such a functional using j p as a variable. The function g() is closely related to the magnetic susceptibility of the homogeneous electron gas. For some electron densities, values of this quantity have been calculated at a correlated level 7 to which a decent analytical approximation can be fitted. The last term of Eq. 5 may be added to any exchange-correlation density-only functional to give the simplest approximation to its currentdependent generalization. Therefore I shall consider in this paper current-density functionals depending locally on,, and, 5 E xc,, F xc,,dr,, 6a F xc,,f xc,g 2, 6b where F xc, is a possibly gradient-corrected exchange-correlation density functional currently in use. Furthermore, only the closed shell case is to be dealt with, which is no restriction when calculating chemical shifts. A finite basis set LCAO scheme will be used exclusively, in which the Kohn Sham orbitals i are expanded in a finite set of atomic basis functions. A computational scheme for the calculation of magnetizabilities within such a LCAO scheme, based on the theory of Vingale et al., 6 has recently been presented by Colwell and Handy. 8 An extension to chemical shifts would be a straight J. Chem. Phys. 102 (7), 15 February /95/102(7)/2806/6/$ American Institute of Physics

2 Christoph van Wüllen: DFT calculation of NMR shifts 2807 forward task. But in spite of the fact that the theory of Vingale et al. is gauge invariant so is e.g., Hartree Fock, actual calculations using finite basis sets are plagued by the so-called gauge origin problem, 9 which is a basis set insaturation problem; in extended systems, conventional basis sets cannot give a proper description of the wave function in the presence of a magnetic field for any choice of the gauge origin. In the next two sections, I will present working equations for the CDFT calculation of chemical shifts using both the IGLO Refs. 10, 11 and GIAO Refs. 12, 13 concepts, which are well-established and rather general methods to overcome the gauge origin problem. II. THE IGLO FORMALISM Let us briefly review the stationarity conditions for the Kohn Sham KS orbitals in the presence of a magnetic field, where the orbitals cannot be chosen real. Indices i, j,k,... will denote doubly occupied orbitals, a,b,c,... virtual orbitals and p,q,r,... any orbitals. Consider now an orbital rotation i i a O 2. 7a This orbital rotation changes the density, the paramagnetic current etc. according to 2 ia 2* ia *O 2, 2 ia 2* ia *O 2, j p j p 2j ia 2*j ia *O 2, 2 ia 2* ia *O 2, with ia i * a, 7b Let the unperturbed MOs be real and satisfy these stationarity conditions. If now a magnetic field B real is switched on, this induces a change in the occupied and virtual orbitals p p p O 2, where the first-order MOs p can be chosen to be purely imaginary. In what follows, a prime denotes the total derivative with respect to at 0. The basic concept of the IGLO method is to localize the unperturbed occupied MOs and to introduce a special ansatz for the first-order change of the MOs, i i i X ik k X ia a, a X aj j. The i are local multiplicative operators defined as i r i 2c rr i B, 9 10a 10b where R i is usually taken to be the charge centroid of the localized MO i. Note that the X pq are purely imaginary. The orthogonality condition for the perturbed MOs yields X ik *X ki i k i k 0, X ia *X ai i i a 0. 10c Since unitary transformation among the occupied orbitals do not change the wave function, a special choice can be made for the X ik, X ik 1 2 k k i i. 10d ia 1 i * a, j ia i 2 i * a a i *, 7c The KS matrix elements will change due to the magnetic field F ia F ia F ia O 2, 11a ia j ia iaj p 2. For the KS energy follows E E2F ia 2*F ia *O 2 7d and the KS matrix elements are given by F ia i h a 2 i a k k F xc ia F xc ia F xc iadr. 7e These quantities can be regarded as matrix elements of a KS operator F, F ia i F a F ai * 7f and the stationarity conditions for the energy simply read F ia 0. 8 F ia i F a i F a i F a, 11b and the yet unknown coefficients X ia can be determined from the equation F ia Putting in the IGLO ansatz Eq. 10 and exploiting the stationarity condition Eq. 8, the first two terms of Eq. 11b can be evaluated such that Eq. 12 reads assuming the virtual orbitals are canonical i i F a X ia F aa F ij X ja F ij j j a i F a Since the Coulomb and exchange-correlation potentials of the unperturbed KS operator are local and thus commute with i, the first term of Eq. 13 can be rewritten as

3 2808 Christoph van Wüllen: DFT calculation of NMR shifts i i F a i h, i a i F i a i h, i a i F p p i a i h, i a i F j j i a. 14 A completeness insertion has been used to reduce some special matrix elements to known quantities. This is a good approximation since the sum over the whole basis reduces itself to a sum over the occupied orbitals because of the stationarity condition. Equation 13 is thus transformed to i h, i a F ij j j i a X ia F aa F ij X ja i F a To evaluate i F a, one has to consider the change of F ia due to h hho 2, O 2, j p j p jo 2, O Fortunately, vanishes for imaginary variations of real wave functions. Then, Y ia F ij X ja X ia F aa 4X jp2g j jp j ia dr0, 19 where the Y ia are matrix elements of an effective perturbation operator Y ia i hh, i a F ij j j i a 1 2g j j R j B c j ia dr. 20 Unlike in the IGLO variant of coupled Hartree Fock, 10 no completeness insertions have been used to simplify exchange-type terms in Eq. 20 since this does not help if the integral is evaluated numerically. Equation 19 can be solved iteratively using techniques which are well known from, say, the coupled Hartree Fock case. Once the X ia have been obtained, the calculation of the nuclear magnetic shielding at the position of all nuclei in the molecule is both straightforward and computationally inexpensive; the Hamiltonian in the presence of a magnetic field B and a magnetic moment of a nucleus at position R N takes the form H,H 0 hh 01 h 11 21a with the gauge origin is at the origin of the coordinate system j2 i 2 j* j j j j j j j * h 1 2ci r B, 4X jp j jp 2i j j j 4X jp j jp 1 c j j R j B, 17a h 01 1 ci rr N rr N 3, 21b j 4X jp j jp 1 c j j R j B. 17b It should be noted that the second term of in Eq. 17b vanishes if all the R j take the same value. This contribution also remains unchanged if a constant vector R is added to all R j. This nontrivial result is the consequence of the special form of the exchange-correlation functional and ensures translational invariance. Since 2 F xc / and 2 F xc / vanish at 0 for the type of exchangecorrelation functional considered here it follows that i F a i h a 2 F xc iadr i h a 2g ia dr. 18 Putting all the pieces together and noting that j p 0 for the unperturbed wave function, one arrives at the IGLO equations in compact notation h 11 1 rr N rb 2c 2 rr N 3. The Hellman Feynman theorem, which holds in this context, 8 gives E,2 i, H, i,. 22 Note that in the IGLO and GIAO schemes, such a simple relation does not hold for E/ because the first-order orbitals are expanded in field-dependent orbitals. Differentiating Eq. 22 again with respect to yields the magnetic shielding, 2 E 2 i h 11 i 4 i h 01 i. 23a 0 Putting in the expression for i Eq. 10 gives, after some reorganization 2 i h 11 h 01, i i 4X ip i h 01 p. 23b

4 Christoph van Wüllen: DFT calculation of NMR shifts 2809 Like in the Hartree Fock case, 10 the first term of Eq. 23b is a local diamagnetic term where the gauge origin associated with the operator h 11 has been moved to the charge centroid of the orbital i for each of the matrix elements. Let us note that the method presented so far reduces to results already known for two special cases; first of all, if one neglects the current-dependent part of the exchangecorrelation functional, one arrives at the IGLO variant of uncoupled DFT as presented by Malkin et al. 4 To get uncoupled equations, it is necessary to transform the Y ia to the canonical orbital basis and to transform the solution back to the localized orbitals. This procedure removes the coupling through off-diagonal KS matrix elements and has been described in detail by the present author in Ref. 11. It also speeds up the convergence in the iterative solution of the IGLO equations if they are coupled by exchange and/or correlation terms as well. The calculation of the response term of a second-order property based upon a stationarity condition can always be formulated in terms of a sum over states SOS formula. 14 This is usually not very helpful in actual calculations since the model excited states to be used and the corresponding excitation energies are given by the eigenvectors and eigenvalues of the electronic hessean. However, in the case of uncoupled DFT, these quantities are known since the hessean is diagonal if canonical orbitals are used, and the eigenvalues are simply differences of orbital energies. Therefore, a very simple SOS approach, which uses single replacement Slater determinants as excited states and orbital energy differences as energy denominators, is equivalent to the uncoupled DFT treatment for the calculation of magnetic properties. Note that this is not supposed to work in the case of real perturbations. Malkin et al. 15 introduced a variant of this simple SOS approach where they applied an ad hoc correction to the diagonal elements of the hessean the energy denominators of the SOS formula based on physical reasoning. They wanted to overcome the deficiencies of the uncoupled DFT treatment without going through the complete CDFT formalism. This improves the results in many cases. But to my knowledge, Malkin et al. have never pointed out that the method is no longer gauge invariant even in the limit of a complete basis set; upon a shift of the gauge origin by a displacement R, i 2c rr B, p pq q, h hh,, h 11 h 11 h 01,, 24a the calculated value for the chemical shift changes according to 4 ia i h 01 a 1 F aaf ii 24b F aa F ii ai, where ai is the correction to the energy denominator. The second special case of the IGLO Eqs. 19 and 20 is obtained if all the j are equal R j R for all j. In this case, all the correction terms and the X ik vanish and Eqs. 19 and 20 become identical to the CDFT equation obtained by Colwell and Handy 8 with R the common gauge origin. Neglecting the current-dependent part of the exchange-correlation functional and using a common gauge origin one eventually arrives at the old schemes of Refs. 1 and 2. III. THE GIAO FORMALISM The GIAO concept is formally very similar to the IGLO approach. The same kind of gauge factors are used, but they are applied to atom-centered basis functions instead of localized MOs. Let p C p 25 be the unperturbed KS orbitals p expanded in atomcentered basis functions Greek indices,,,,... will be used to denote basis functions. Within the GIAO concept, the first-order change of the MOs is given by p C p X pq q,, i 2c rr B, 26a R is chosen to be the position where the basis function is centered, and the X pq are purely imaginary. The orthogonality relation for the perturbed MOs yields X* pq X pq C p C q 0 26b and the X ik can be chosen X ik 1 2C k C i. 26c The similarity to the IGLO case becomes obvious if one compares Eqs. 10 and 26. To get the GIAO equation for the X ia, we again have to evaluate F ia Eq. 11b. The first two terms give assuming all unperturbed orbitals are canonical i F a i F a ) X ia *F aa X ai F ii C i C a F F X ia F ii F aa C i C a F ii C i C a F F. 27 The last term of Eq. 27 involves the well-known GIAO one- and two-electron integrals as well as an integral involving the unperturbed exchange-correlation potential

5 2810 Christoph van Wüllen: DFT calculation of NMR shifts F F 4X jp j jp 2iC j C j h h 2C k C k F xc 4X jp j jp 1 c C jc j R R B r R B, 30a 1 F xc dr. 28 It is sometimes convenient to rewrite the Coulomb part of this expression 2C k C k 2C k C k 2C k C k. 29 Equation 28 is similar to the Hartree Fock case, with the exchange-type contribution from the GIAO two-electron integrals replaced by an integral involving F xc. The evaluation of i F a follows the same lines as in the IGLO case. We note that assuming that the unperturbed orbitals are real 0 and j p 2 i 2 j* j j j j j j j * j p 4X jp j jp 1 c C jc j R R B r R B. 30b Again it can be seen that the second term of Eq. 30b vanishes if R R for all, and that translational invariance is ensured. The GIAO equations finally read Y ia X ia F ii F aa 4X j jp jp2g with j ia dr0 31 Y ia C i C a hh h 2C k C k F xc 1 F xc dr 1 c C jc j 2g j ia R R B r R B drf ii. 32 Most of the terms occur in the GIAO variant of couple Hartree Fock as well, except for the integrals, which replace exchange-type terms. Equation 31 can be solved using standard methods, and the chemical shifts are calculated by putting in the GIAO expression for i Eq. 26a into Eq. 23a, 2 i h 11 i 4 i h 01 i 2C i C i h 11 h 01 h 01 4X ip i h 01 p. 33 If one neglects the current-dependent part of the exchangecorrelation functional, Eqs. 31, 32 describe the GIAO variant of the uncoupled DFT scheme. Results of this method have been reported by Friedrich et al., 3 but no computational details were given. In the one-center case for all,, the GIAO formalism is equivalent to using a common gauge origin at that center. IV. CONCLUDING REMARKS When comparing the computational effort associated with the IGLO and GIAO schemes presented in the preceding sections, one has to distinguish between uncoupled DFT and the full CDFT treatment. Within a DFT implementation which uses two-electron integrals for the Coulomb part and a numerical integration for the exchange-correlation energy such as e.g. described by the present author 16 and many others, uncoupled DFT with GIAOs has the drawback that the GIAO two-electron integrals have to be calculated and that a numerical integration is necessary, whereas both of these tasks are eliminated in the IGLO approach by the use of a completeness insertion. Thus the uncoupled DFT-IGLO method requires only a small fraction of the time involved in the computation of the Kohn Sham orbitals itself. In the full CDFT treatment on the other hand, this difference does not matter that much since the iterative solution of the CDFT equations requires several numerical integrations in either

6 Christoph van Wüllen: DFT calculation of NMR shifts 2811 approach; doing a common gauge origin CDFT calculation or an IGLO or GIAO variant thereof will require comparable computational effert unless the evaluation of the GIAO twoelectron integrals which is avoided in IGLO dominates. It should be noted that both schemes can be implemented without using complex arithmetics since all quantities are either real or purely imaginary. 1 W. Bieger, G. Seifert, H. Eschrig, and G. Grossmann, Chem. Phys. Lett. 115, D. G. Freier, R. F. Fenske, and Y. Xiao-Zeng, J. Chem. Phys. 83, K. Friedrich, G. Seifert, and G. Grossmann, Z. Phys. D 17, V. G. Malkin, O. L. Malkina, and D. R. Salahub, Chem. Phys. Lett. 204, A. K. Rajagopal and J. Callaway, Phys. Rev. B 7, a G. Vingale and M. Rasolt, Phys. Rev. Lett. 59, ; b G. Vingale, M. Rasolt, and D. J. W. Geldard, Adv. Quantum Chem. 21, G. Vingale, M. Rasolt, and D. J. W. Geldard, Phys. Rev. B 37, S. M. Colwell and N. C. Handy, Chem. Phys. Lett. 217, W. Kutzelnigg, J. Mol. Struct. Theorchem 202, M. Schindler and W. Kutzelnigg, J. Chem. Phys. 76, Ch. van Wüllen and W. Kutzelnigg, in METECC-94, edited by E. Clementi STEF, Cagliari, 1993, Vol. B, p R. Ditchfield, Mol. Phys. 27, K. Wolinski, J. F. Hinton, and P. Pulay, J. Am. Chem. Soc. 112, W. Kutzelnigg, Theor. Chim. Acta. 83, V. G. Malkin, O. L. Malkina, M. E. Casida, and D. R. Salahub, J. Am. Chem. Soc. 116, Ch. van Wüllen, Chem. Phys. Lett. 219,