Electronic g-tensors obtained with the mean-field spin–orbit Hamiltonian

17 January 2002

Chemical Physics Letters 351 (2002) 424–430 www.elsevier.com/locate/cplett

Electronic g-tensors obtained with the mean-field spin–orbit Hamiltonian Olav Vahtras b

a,*

, Maria Engstr€ om b, Bernd Schimmelpfennig

c

a Center for Parallel Computers, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Department of Physics and Measurement Technology, Link€oping University, SE-581 83 Link€oping, Sweden c Department of Theoretical Chemistry, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Received 27 November 2000; in final form 22 November 2001

Abstract The accuracy of the atomic mean-field approximation (AMFI) to the spin–orbit interaction Hamiltonian is evaluated for the paramagnetic contribution to electronic g-tensors. A variety of different substances are tested for this purpose for restricted open-shell Hartree–Fock and multi-configuration self-consistent field wave functions. In most cases the introduced error is significantly lower than errors of wave function parameterization. Considering the substantial computational savings, it is argued that the mean-field approximation is warranted given the resolution available in current electron spin resonance experiments. Ó 2002 Elsevier Science B.V. All rights reserved.

1. Introduction The importance of spin–orbit interaction is increasingly being recognized in a variety of catalytic and spectroscopic processes for molecular species containing elements across the whole periodic table, also within the first rows, see [1]. Although it is well motivated to use the spin–orbit interaction in a perturbative fashion, the calculations still become costly even for light elements. The computational cost is due to low symmetry of the full spin–orbit operator, which includes two-electron terms with contributions from spin–own–orbit and spin–other–orbit interactions. In traditional semi-

*

Corresponding author. Fax: +46-8-24-77-84. E-mail address: [email protected] (O. Vahtras).

empirical work, the spin–orbit integrals are approximated by including the one-electron part of the operator only and/or considering the onecenter contributions to the spin–orbit operator [2,3]. While the latter approximation often has been found well motivated, the complete neglect of the two-electron terms can fail at times [4]. The two-electron interaction can be treated in an average way, screening the nuclear charge for the effective one-electron spin–orbit interaction, by an atomic mean-field approximation (AMFI) to the spin–orbit operator. It has been argued that this mean-field approximation forms a good balance between efficiency and accuracy [5,6]. In fact, the performance of AMFI has recently been demonstrated in the calculation of several effects where spin–orbit coupling is important, such as spin– orbit splittings [5,6], spin–orbit induced effects on

0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 4 3 3 - 6

O. Vahtras et al. / Chemical Physics Letters 351 (2002) 424–430

the nuclear shieldings [7], singlet–triplet excitation spectra [8] and phosphorescence radiative lifetimes [9]. The good performance of AMFI in terms of efficiency and accuracy makes it attractive to test AMFI for other vital response properties containing the spin–orbit operator as a perturber. The paramagnetic contribution to electronic g-tensors is derived from the second-order energy correction associated with the spin–orbit interaction and the orbital Zeeman effect [10]. Recently, g-tensors have been addressed at the DFT level [11,12]. Several of the molecules of this Letter were also studied by Schreckenbach and Ziegler [11], and in the the work by Malkina et al. [12] the mean-field spin–orbit Hamiltonian was used in the context of g-tensors for the first time. We have previously studied g-tensors by (linear) response theory for self-consistent field (SCF) and multiconfiguration self-consistent field (MCSCF) reference states using the full two-electron spin–orbit Hamiltonian [13–16] Owing to its fundamental role in electron paramagnetic resonance (EPR) spectroscopy, it is motivated to widen the applicability of ab initio response theory to more extended species, by introducing the mean-field Hamiltonian also in this context. The purpose of this Letter is to take a step in that direction.

425

where i, j and K refer to the electrons and the nuclei, lij define the angular respectively. Thus ~ liK and ~ momenta of electron i with respect to the position of nucleus K or electron j, ~ rij are the corresponding position vectors, and~ si the electron spin operators. a is the fine-structure constant. The lack of particle symmetry in the spin–orbit operator increases significantly the cost of computing and storing integrals compared to calculations involving electrostatic two-electron repulsion integrals. The two-electron contribution is opposite in sign and partly cancels the one-electron contribution. This has led to approximations where the two-electron part is neglected and compensated for by an effective nuclear charge. In second row atoms the two-electron spin–orbit effects are often of the same order in magnitude as the one-electron part, and it is not obvious that such a screening gives reliable approximations. On the other hand, one-electron spin–orbit effects grows with nuclear charge as Z 4 , and it is obvious that such an approximation will improve with heavier elements in the system. Heß et al. [5] describe a Fock-type generalization of the spin–orbit operator with account of the interactions between single valence–valence excited state determinants. This leads to Hijsomf ¼ hijHso ð1Þjji

2. Theory

þ

The electronic g-tensor parameterizes the interactions between electron spin ~ S and external magnetic fields ~ B as described by the spin Hamiltonian H ¼ lB~ S g ~ B:

mforbs: X

occðMÞ½hiMa jHso ð2ÞjjMa i

M

þ hiMb jHso ð2ÞjjMb i  hiMa jHso ð2ÞjMa ji  hiMb jHso ð2ÞjMb ji  hMa ijHso ð2ÞjjMa i  hMb ijHso ð2ÞjjMb i;

ð1Þ

ð3Þ

The g-shift Dg is calculated as a sum of Breit–Pauli terms, i.e. relativistic mass contribution, gauge corrections and spin–orbit/orbital–Zeeman contributions [10]. The full Breit–Pauli spin–orbit operator contains both one- and two-electron parts,

where occðMÞ denotes the occupation number of orbital M, i and j spin–orbitals, and Ms the partially occupied orbitals with which the electronic charge distribution interacts. The computational cost is reduced further by neglecting all multicenter contributions to both the one- and twoelectron spin–orbit interactions [6]. The screening of the nuclear charge and the calculations of the spin–orbit integrals is thereby reduced to individual atoms with full atomic symmetry imposed (separation of radial and angular parts).

Hso ¼ Hso ð1Þ þ Hso ð2Þ # " X~ a2 X ~ liK lij ZK 3 ~ ð~ si þ 2~ sj Þ ; ¼ si  2 iK riK rij3 ij

ð2Þ

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3. Computational details

4. Results and discussion

All calculations of the atomic mean-field spin– orbit integrals have been performed with the AMFI program of Schimmelpfennig [17], which is interfaced to the DA L T O N quantum chemistry program [18] used for the present g-tensor calculations. The geometries of CH3 , SiH3 , GeH3 , ClO2 , ClSO and ClSS were obtained with the B3LYP/ 6-31G method using NWC H E M [19] and GA U S S I A N 98 [20] and the atomic natural orbital (ANO) basis set [21–23] was used in the gtensor calculations. SCF is here used synonymously for restricted open-shell Hartree–Fock wave functions. MCSCF wave functions were constructed from five electrons in five active orbitals (CH3 , SiH3 , GeH3 ) and seven electrons in seven active orbitals (ClO2 , ClSO, ClSS). The g-tensors of the substituted benzene radicals were calculated according to earlier work [15], using cc-pVDZ basis set [24].

The mean-field approximation was originally devised for systems containing transition-metal compounds [5], but it is still relevant to test it on organic systems containing first and second row elements. The reason is that the mean-field approximation has produced encouraging results at this level for other properties [9]. In Tables 1–3, g-tensors, which include the free-electron term and the paramagnetic shifts, are collected for the full spin–orbit operator as well as the mean-field approximation. The XH3 radicals (Table 1) posses trigonal symmetry; CH3 is planar while both SiH3 and GeH3 are pyramidal in shape. In a small light molecule as CH3 , the mean-field approximation produces relatively large errors. In Table 1, it is shown that the relative error in the paramagnetic shift for both the SCF and MCSCF calculations is of the order of 20%, which is much larger than the correlation effects. However, this error

Table 1 g-tensors for XH3 (X ¼ C, Si, Ge) using SCF and MCSCF reference wave functions, calculated within mean-field spin–orbit approximation (AMFI) as well as with the full two-electron spin–orbit Breit–Pauli operator (BP) Molecule

Method

gij

AMFI

BP

D

Exp.

CH3

SCF

gk g? giso gk g? giso gk g? giso gk g? giso gk g? giso gk g? giso

2.00232 2.00279 2.00263 2.00232 2.00277 2.00262 2.00232 2.00343 2.00306 2.00231 2.00362 2.00317 2.00232 2.01248 2.01248 2.00200 2.01443 2.01028

2.00237 2.00289 2.00272 2.00238 2.00288 2.00271 2.00236 2.00338 2.00304 2.00231 2.00358 2.00316 2.00234 2.01259 2.01259 2.00204 2.01454 2.01037

)48 )105 )86 )60 )155 )89 )41 43 15 )41 42 14 )49 )113 )85 )36 )109 )85

2.0024a 2.0027a

MCSCF

SiH3

SCF

MCSCF

GeH3

SCF

MCSCF

D is the approximation error (ppm). a Ref [25]. b Ref [26].

2.003b 2.007b

2.003b 2.017b

O. Vahtras et al. / Chemical Physics Letters 351 (2002) 424–430

427

Table 2 g-tensor for ClXX radicals (see caption of Table 1) Molecule

Method

gij

AMFI

BP

D

ClO2

SCF

g1 g2 g3 giso g1 g2 g3 giso g3 g2 g1 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso

2.00232 2.02899 2.02321 2.01817 2.00235 2.01601 2.02001 2.01279 2.00232 2.01565 2.00911 2.00903 2.00191 2.01579 2.00922 2.00897 2.00231 2.02244 2.05036 2.02504 2.00221 2.02101 2.04224 2.02182

2.00248 2.02913 2.02336 2.01832 2.00253 2.01605 2.02012 2.01290 2.00245 2.01567 2.00919 2.00910 2.00205 2.01580 2.00930 2.00905 2.00239 2.02241 2.05048 2.02509 2.00228 2.02099 2.04233 2.02187

)166 )141 )148 )152 )187 )34 )112 )111 )137 )14 )80 )77 )138 )14 )83 )78 )79 28 )113 )55 )74 19 )85 )47

MCSCF

ClSO

SCF

MCSCF

ClSS

SCF

MCSCF

Exp.

2.0036a 2.0183a 2.0088a

2.000b 2.017b 2.008b

2.0019c 2.0225c 2.0384c

a

Ref. [27]. Ref. [28]. c Ref. [29]. b

is insignificant when comparing the total g-tensor with experiment. The g-shifts for both the GeH3 and SiH3 show the opposite trend. That is, the correlation effects are relatively large, but the mean-field errors are in the order of 1% for the isotropic g-shifts. There are individual elements with substantially larger relative errors, such as the SCF gk components. However, these errors are small on an absolute scale. The absolute error should be compared with experimental errors which can typically be 50 ppm in high field EPR where the g-tensor is well resolved. Furthermore, the experimental errors are even higher in standard low-field (9.5 GHz) EPR. Other individual elements have large errors on a relative scale but small on an absolute scale. The ClO2 radical is a triatomic radical of ABA type (C2v symmetry) with substantial spin density localized to the chlorine atom. Both ClSO and ClSS are ABC type radicals (Cs symmetry) and

they have almost zero spin density on the chlorine atom. In Table 2 comparisons between the g-tensors of the triatomic radicals calculated from SCF and MCSCF reference states are displayed. For all entries the g1 component refers to the out-of-plane direction and the g3 direction bisects the bond angle, in the first case the C2v symmetry axis. The differences in the isotropic g-values between the AMFI and standard methods were on average Dgav ¼ 86 ppm. However, the difference between SCF and MCSCF wave functions are more pronounced. For example, including correlation to ClSS changes the full two-electron isotropic g-shift by 14%, whereas the mean-field error is only )0.2% for both wave functions. In a previous study the g-tensors of a series of substituted benzene radicals were calculated [15]. We refer to this Letter for a more detailed discussion as well as references to experiments. In this

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O. Vahtras et al. / Chemical Physics Letters 351 (2002) 424–430

Table 3 g-tensors for substituted benzene radicals (see caption of Table 1) Molecule

gij

AMFI

BP

D

Benzene cation

g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso g1 g2 g3 giso

2.00232 2.00269 2.00225 2.00242 2.00232 2.00242 2.00296 2.00257 2.00232 2.00306 2.00380 2.00306 2.00232 2.00394 2.00361 2.00329 2.00232 2.00338 2.00703 2.00424 2.00232 2.00426 2.00425 2.00361 2.00232 2.00523 2.00619 2.00458 2.00232 2.00518 2.00614 2.00455

2.00252 2.00289 2.00246 2.00262 2.00262 2.00266 2.00315 2.00281 2.00253 2.00306 2.00395 2.00318 2.00251 2.00396 2.00377 2.00341 2.00243 2.00341 2.00720 2.00435 2.00253 2.00444 2.00441 2.00379 2.00259 2.00550 2.00634 2.00481 2.00260 2.00544 2.00629 2.00478

)201 )203 )212 )206 )305 )239 )189 )245 )208 0 )144 )117 )194 )21 )163 )126 )111 )38 )173 )108 )211 )176 )159 )182 )272 )271 )146 )230 )276 )255 )157 )229

Benzene anion

Dihydropyrazine cation

Hydropyrazine

Aniline

Benzoquinone anion

Nitropyridine anion

Nitrobenzene anion

Letter, these calculations were reproduced using the AMFI. What is noticeable is that the outof-plane component (g1 ) almost vanishes in the mean-field approximation. Again, this is due to a combination of wave function parameterization and one-electron approximation. It has serious consequences for the benzene cation, C6 Hþ 6 , and anion, C6 H , where the in-plane components are 6 rather small giving a total isotropic shift that is reduced by two thirds in the former case and by a half in the latter. The in-plane components in the other substituted radicals are less sensitive to the approximation, but the overall relative error is still in the order of 10%. The large relative error for individual tensor components is caused by a combination of the

parameterization method and a one-electron approximation (not AMFI in particular) and using triplet operators in a singlet context: if a variational wave function is parameterized by k and the unperturbed energy is E, this g-tensor contribution can be written as  1 oLi o2 E oHSO;j gij ¼  : ð4Þ ok ok2 ok In the conventional sum-over-state expression, the intermediate excited states are restricted to states with the same multiplicity as the reference state, L being a singlet operator. Therefore, we describe the response of the wave function in terms of singlet excitation operators. With an exponential singlet parameterization (conventional RPA), a one-elec-

O. Vahtras et al. / Chemical Physics Letters 351 (2002) 424–430

tron triplet operator has only a few non-zero gradient elements, those describing rotations into and out of the open shell orbital. For a hypothetical case where there are no orbitals in the symmetry that matches the symmetry of the open shell orbital and the operator, a one-electron contribution is identically zero. This effect can be seen for several molecules in Tables 1–3: in a given symmetry there may be few orbital rotations that contribute to the one-electron spin–orbit gradient. If the one-electron matrix elements associated with the open-shell orbital are small, it may happen that the two-electron spin–orbit contribution is substantially larger than the one-electron contribution. The issue of time savings for an approximation like AMFI should evidently be considered. The relevant parts of the calculation to compare are the integral timings and the response timings (the time to calculate the g-factors given all integrals and wave function parameters). The observed differences in integral timings reflect that in the full twoelectron case, all two-electron spin–orbit integrals are calculated and saved on file, whereas in the mean-field case, only one-center integrals are considered. The speedup is of the order a factor of 50 for the small molecules and up to a factor of 1000 for the largest. The differences in the response timings are due to the construction of the righthand side (the spin–orbit gradient) that is used in the linear response equations to be solved. The construction of the right-hand side for a twoelectron operator is comparable to a normal (MC)SCF iteration, and the observed speed-up of an order of 10 for the largest molecules. Last and not least, the disk space is another, severe, limiting factor for full spin–orbit calculations. To get around this bottle-neck it would be possible to calculate the integrals in a direct fashion in SCF, i.e. to throw them away as they have been added to the spin–orbit Fock matrix. In principle, this would be done without performance loss, since the integrals are only needed once. However, correlation is often important for spin– orbit effects, and an integral direct MCSCF scheme is not as straight forward as in the SCF case, and would in any case lead to substantially longer execution times.

429

The results accounted for above show that the deviations in the g-values caused by the atomic mean-field approximation are smaller than deviations caused by other approximations, such as choice of wave function and molecular geometry. As expected, calculations using the AMFI approximation are time saving. The saving is made in the integral calculation rather than any other part of the code. The net reduction in computational time is or the order of a factor of 10 (not considering the wave function calculation).

5. Concluding remarks This study is the latest in a series of demonstrations of the applicability of the mean-field approximation in estimating the contribution of spin–orbit matrix elements in different properties and processes [5–7]. It is our conclusion that the atomic mean-field approximation is efficient in reducing the computational cost in g-tensor calculations. Substantial time-savings are first of all made in the calculation of the integrals, because only one-center spin–orbit integrals are considered in the mean-field approximation. The disk space requirement is another limiting factor, although not essential for the systems in this study. Even for slightly larger systems this will, however, be a bottle-neck, and it will become necessary to apply integral direct methods. With our current implementation this is restricted to SCF wave functions. The accuracy using AMFI for g-tensors is remarkable for systems containing atoms beyond the second row. In these cases, we find that the errors are completely negligible to other errors which arise from the wave function parameterization. In particular at the SCF level the correlation error is an order of magnitude larger than the errors associated with the mean-field approximation.

Acknowledgements This work was financed by the Swedish Foundation for Strategic Research via the Forum Sci-

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O. Vahtras et al. / Chemical Physics Letters 351 (2002) 424–430

entum graduate school. Computer time was supported by PDC, Center for Parallel Computers, at KTH, Stockholm.

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