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Calculation of electric properties using regular approximations
to relativistic effects: The polarizabilities of RuO 4 , OsO 4 ,
and HsO 4 „ ZÄ 108 …
Michael Filatova)^ and Dieter Cremer Department of Theoretical Chemistry, Go¨teborg University, Reutersgatan 2, S-41320 Go¨teborg, Sweden ~Received 21 March 2003; accepted 16 April 2003! Analytic expressions for the derivatives of the total molecular energy with respect to external electric field are derived within the regular approximation to the full four-component relativistic Hamiltonian and presented in matrix form suitable for implementation in standard quantum-chemical codes. Results of benchmark calculations using the infinite-order regular approximation with modified metric method are presented and discussed. The static electric dipole polarizabilities of group VIII metal tetroxides MO 4 for M 5 Ru, Os, Hs ~Z 5108! are studied with the help of second-order Møller–Plesset perturbation theory using the infinite-order regular approximation with modified metric Hamiltonian. The polarizabilities obtained vary in the sequence RuO 4 .OsO 4 .HsO 4 , which is different from those obtained in other studies. However, it is in line with calculated 1 T 2 ←^1 A 1 excitation energies of the group VIII tetroxides, which provide a measure for the magnitude of their polarizabilities. © 2003 American Institute of Physics. @DOI: 10.1063/1.1580473#
I. INTRODUCTION
The electric dipole polarizability is a measure of the dis- tortion of the overall atomic or molecular charge distribution by an external electric field.^1 The polarizability is a ubiqui- tous parameter that appears in many formulas for low-energy processes involving the valence electrons of atoms and molecules.^2 Knowledge of atomic and molecular polarizabil- ity is important in many areas of computational chemistry ranging from electron^3 and vibrational^4 spectroscopy to mo- lecular modeling,^5 drug design,^6 and nano-technology.^7 While the bulk of previous quantum-chemical calcula- tions of polarizabilities focuses on compounds containing just light elements, there is now a growing interest in obtain- ing the electric response properties for inorganic and organo- metallic compounds containing heavy elements. Heavy atom compounds can only be correctly described by quantum- chemical methods including relativistic corrections. The core electrons of a heavy atom experience the largest effects of relativity, however, they have vanishingly small influence on to polarizability and other electric response properties of a molecule. Nevertheless these properties experience indirect relativistic effects via contraction or decontraction of valence and sub-valence electrons caused by relativity,8 –10^ which can be significant. Thus, there is currently a substantial interest in developing computational methods capable of handling rela- tivistic effects on molecular response properties.11– Although fully relativistic four-component calculations have been carried out12,13,15,16^ for a number of atoms and small molecules, the computational complexity of the rigor- ous four-component formalism based on the Dirac Hamiltonian^17 fuels the quest for simple yet reliable quasire- lativistic methods. The methods based on the Douglas–Kroll ~DK! transformation^18 as well as on a regular expansion19,
of the Dirac Hamiltonian have been established in recent years as cost effective alternatives to the full four-component formalism. Recently, we have developed21,22^ a new algorithm for calculation of the matrix elements of the Hamiltonian within regular approximation ~RA! approach, which can easily be applied in the context of basis set ab initio calculations. At variance with the Douglas–Kroll–Hess approach,^23 the new computational procedure is stable with respect to basis set reduction and produces reasonable results even with small basis sets.^22 For the purpose of uprooting the erroneous gauge dependence of the eigenvalues of the regular Hamiltonian,19,20^ a simple modification of the wave function metric within the infinite-order regular approximation ~IORA!^20 method was suggested, which led to the IORA method with modified metric ~IORAmm!.21,22^ The IORAmm method21,22^ was successfully applied to study compounds of heavy elements within the context of ab initio wave function theory.21,22,24, The purpose of the present paper is to extend the formal- ism presented in our earlier works21,22^ so that it can be ap- plied to the calculation of electric response properties. This development should provide the possibility of calculating relativistically corrected dipole moments, polarizabilities or hyperpolarizabilities of large molecules using just standard wave function ab initio techniques. It should be noted that previously the relativistic methods based on the regular ap- proximation were used^26 almost exclusively within the con- text of density functional theory ~DFT!,^27 which often does not achieve a satisfactory level of accuracy when calculating electric response properties.^28 Results of the current work will be presented in the fol- lowing way. In Chapter II the IORAmm formalism is de- scribed briefly along with the formulas necessary to calculate a!Electronic mail: filatov@theoc.gu.se the electric dipole moments and electric dipole polarizabil-
JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 3 15 JULY 2003
0021-9606/2003/119(3)/1412/9/$20.00 1412 © 2003 American Institute of Physics
ities. Chapter III presents results of benchmark calculations performed for various atomic and molecular systems, for which the effect of relativity on static electric dipole polar- izabilities varies from very small ~noble gas atoms! to very large ~mercury atom!. The formalism developed is applied in Chapter IV to study electric dipole polarizabilities of group VIII metal tetroxides MO 4 for M 5 Ru, Os, Hs ~Z 5108 !. There is substantial interest in studying the properties of compounds containing super-heavy elements such as has- sium ~Hs!.29–33^ The results of the present work provide a basis for a reassessment of the values of the polarizabilities of RuO 4 and HsO 4 calculated recently^30 at the level of rela- tivistic DFT calculations.^34
II. THEORY OF THE RELATIVISTIC DESCRIPTION OF ELECTRIC PROPERTIES The energy of a molecule in a static uniform electric field F can be expanded as in Eq. ~ 1! in terms of the field strength
E ~ F! 5 E ~ 0! 1
] E ~ F!
] F
U
F 50
• F 1
F •
]^2 E ~ F!
] F ] F
U
F 50
• F 1 .... ~ 1!
From this expansion the dipole moment is defined as in Eq. ~ 2!
m 52
] E ~ F!
] F U F 50
and the polarizability tensor as in Eq. ~ 3!
a 52
]^2 E ~ F!
] F ] F U F 50
For convenience the experimentally measured polarizability is characterized by two parameters, isotropic polarizability ~ 4! a ¯ (^) 5 13 ~a xx 1 a y y 1 a zz !, ~ 4!
and anisotropic polarizability ~ 5! g^25 12 ~~a xx 2 a y y!^21 ~a y y 2 a zz!^21 ~a zz 2 a xx!^2
16 ~a xy^2 1 a^2 yz^1 a zx^2 !!, ~ 5! which are invariant with respect to an orthogonal coordinate transformation. A. The IORAmm methodology The matrix IORAmm ~or IORA! SCF equations in the one-electron (1 ¯e ) scalar-relativistic ~SR! approximation^35 are given in Eq. ~ 6 !21,
~~ S 1/2!†~ U^2 1/2!†~ V n 1 T 1 W !~ U^2 1/2!~ S 1/2! 1 J 2 K! C i
5 SC i e i , ~ 6! where V n is the matrix of the electron–nuclear attraction
integrals ^x mu V n u x n & ( xm : basis functions; V n : Electron–
nuclear attraction potential! and the matrix U is given by Eq. ~ 7!
U 5 S 1
2 mc^2
~ T 1 a W 1 b WT^21 W !, ~ 7!
~parameters a and b for IORA: a 5 2, b 5 1; for IORAmm: a 5 3/2, b 5 1/2). Matrix T is the usual kinetic energy matrix, J and K the Coulomb and exchange matrices, S the overlap matrix, C i the column-vector of expansion coefficients, and e i the eigenvalue of the orbital c i. The matrix W in Eqs. ~ 6! and ~ 7! is the solution of Eq. ~ 8! W 5 W 01 W 0 T^21 W , ~ 8! which is given in Eq. ~ 9!
W^215 W 0212 T^21. ~ 9! The matrix W 0 in Eqs. ~ 8! and ~ 9! is calculated in the 1 ¯e^ SR approximation according to Eq. ~ 10!
W 05
4 m^2 c^2
^xmu p V^ n • p u^ x^ n &.^ ~^10!
Note, that only the electron–nuclear attraction potential V (^) n is used in Eq. ~ 10! and the electron–electron repulsion is treated nonrelativistically. This implies an assumption that the Foldy–Wouthuysen transformation^36 commutes with the electron–electron repulsion operator.^35 The solution of the IORAmm/IORA equation ~ 6! approximates the true relativ- istic wave function in the Foldy–Wouthuysen representation.^20 The total IORAmm/IORA SCF energy is given by Eq. ~ 11!
E SCF 5 tr ~ 12 P ~ H 1 F !!
5 tr ~ P ~~ S 1/2!†~ U^2 1/2!†~ V n 1 T 1 W !~ U^2 1/2!~ S 1/2!
1 12 ~ J 2 K !!!, ~ 11! where P is the density matrix in the basis of functions xm
P 5 CnC †^. ~ 12! In Eq. ~ 12 !, n is the diagonal matrix of orbital occupation numbers. Eq. ~ 11! differs from the corresponding nonrelativ- istic expression only in the use of the relativistically cor- rected one-electron Hamiltonian H 1 ¯e for the calculation of the one-electron part to the total SCF energy. When the molecular integrals in Eq. ~ 11! depend on an external parameter l, the first derivative of the self- consistent field ~SCF! total energy with respect to the param- eter l is given in Eq. ~ 13!^37
] E ]l
5 tr S P S
]
]l
H D D 1 tr S WS
]
]l
S D D
tr S P S
] 8
]l
~ J^2 K! D D.^ ~^13!
Here W is the energy-weighted density matrix W 5 Cn e C †^ , ~ 14! where e is the diagonal matrix of orbital energies. The prime at ]/ ]l implies that only molecular two-electron integrals rather than density matrix elements ~or orbital coefficients! have to be differentiated. In an external electric field F with components F a ( a 5 x , y , z ), the potential V n in Eq. ~ 11! modifies to
J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 Polarizabilities of RuO 4 , OsO 4 , and HsO 4 1413
With the help of Eqs. ~ 19 !–~ 22! it can be demonstrated that the third term on the right hand side ~r.h.s.! of Eq. ~ 24! is of the order 1/ c^4 , the fourth and fifth terms both contain con- tributions of order 1/ c^4 and 1/ c^6 , the sixth and seventh terms are of orders 1/ c^6 and 1/ c^8 , respectively. Differentiating Eq. ~ 19! with respect to a field component F b Eq. ~ 25! for the second derivative of the matrix H is derived
]^2 H
] F b ] F a
16 m^4 c^4
~ WW 021 Q a ~ W 021 WW 0212 W 021!
3 Q b W 021 W 1 WW 021 Q b ~ W 021 WW 0212 W 021!
3 Q a W 021 W !, ~ 25!
which shows that the second term on the r.h.s. of Eq. ~ 24! is of the order 1/ c^4. Because the interaction with an external electric field does not contain singular operators, the contri- butions of all terms except of the first one in Eq. ~ 24! can safely be neglected. Furthermore, the first term contains contributions of both zeroth- and second-order in 1/ c @see Eq. ~ 19 !# and only the zeroth-order terms make sizable con- tributions to the polarizability tensor. The second order terms (;1/ c^2 ), correspond to the picture change in the dipole mo- ment operator and can be neglected as was demonstrated in the literature.11,38^ Thus, the expression for the polarizability tensor reduces to Eq. ~ 26!
a ba 52 tr S S
] P
] F b D
~ G † R a G! D, ~ 26!
which contains the renormalized dipole moment integrals and derivatives of the density matrix P calculated with the help of the usual coupled-perturbed ~CP! equations^39 using the electric field as perturbation. Note, that the CP equations employ the renormalized dipole moment integrals as well. Equation ~ 26! could also be obtained by taking the de- rivative of the dipole moment vector given in Eq. ~ 23! and neglecting contributions of the order 1/ c^4 resulting from the differentiation of matrix G. However, we prefer to present the explicit derivation as given in Eq. ~ 24! because this is needed when, e.g., the quadrupole shielding factor @to be used in nuclear magnetic resonance ~NMR!# has to be calcu- lated. While Eqs. ~ 23! and ~ 26! are applicable to the calcu- lation of dipole moment and polarizability, the effects of a picture change neglected in these equations can become more important for other electric properties such as the elec- tric field gradient ~quadrupole shielding factor! at the nucleus.^38 In the latter case, at least the terms of the order 1/ c^2 should be retained in expressions for the derivatives of the total energy with respect to the nuclear quadrupole mo- ment. However, as Eq. ~ 19! shows, these additional terms can easily be evaluated using the standard nonrelativistic multipole integrals.
III. IMPLEMENTATION, BENCHMARK TESTS, AND CALCULATIONAL DETAILS
The computational scheme described in the previous section was programmed and implemented into the CO- LOGNE2003 suite of quantum-chemical programs.^40 The implementation is straightforward, because it requires only
the renormalization of the dipole molecular integrals on the quasi-relativistic metric ~ 17 !. Since only the one-electron part of the molecular Hamiltonian is modified in the IORAmm calculations, the cost of these calculations is es- sentially the same as the cost of the corresponding nonrela- tivistic calculations. Furthermore, the formalism described in the preceding section can be applied at the Hartree–Fock ~HF! level as well as correlation corrected levels of theory such as Møller–Plesset many body perturbation theory, coupled cluster theory, etc. A. Benchmark calculations For the purpose of testing the effectiveness of the proce- dure described, IORAmm benchmark calculations of the static electric dipole polarizability were carried out for noble gas atoms He through Xe, for the mercury atom, and for hydrogen halides FH through AtH ~Z 585! using a HF wave function. For comparison, nonrelativistic HF calculations were carried out for the same atoms and molecules. For the calculation of the static electric dipole polariz- abilities of the noble gas atoms the uncontracted TZV basis set of Ahlrichs was employed.^41 The original TZV basis set was amended by two sets of diffuse functions and a set of polarizing functions as described by Klopper et al.^42 Hence, the basis sets used for calculation of the polarizability of noble gas atoms He through Xe comprise the following functions: He(7 s 4 p ), Ne(13 s 8 p 4 d ), Ar(16 s 11 p 5 d ), Kr(19 s 15 p 8 d 3 f ), and Xe(21 s 17 p 11 d 6 f ). Only cartesian Gaussian-type primitive functions are employed. The results of IORAmm/HF and nonrelativistic HF cal- culations are presented in Table I where they are compared with the results of the Dirac–Fock ~DF!, 17,43^ direct perturba- tion theory ~DPT!^44 and scaled ZORA^19 calculations taken from the work of Klopper et al.^42 The literature data were obtained with the same basis sets, but using numeric differ- entiation of the total atomic energy with respect to the exter- nal electric field.^42 The comparison reveals that the IORAmm results reproduce closely the results of DF and other quasirelativistic calculations. The effect of a picture change in the dipole moment operators does not show up as is obvious from the comparison of the IORAmm results where this effect is neglected, with the DPT and scaled ZORA results, which take this effect into account. The overall effect of relativity on the polarizabilities of the noble gas atoms is weak, which is consistent with minor relativistic effects observed for the atomic valence p -orbitals.8 –10^ Much stronger influence of relativity should be expected in situations where valence s -orbitals, which ex- perience a substantial relativistic shift, are distorted by an external electric field as in the case of the mercury atom. The enormous relativistic effect on the static electric dipole po- larizability of mercury is well established.45– 47^ According to numeric DF calculations,^45 the polarizability of mercury changes from 80 to 43 bohr^3 upon inclusion of relativity. In the last line of Table I, the results of nonrelativistic HF and IORAmm/HF calculations are compared with the literature data. 45– 47^ The @ 17 s 14 p 9 d 8 f # basis set on mercury employed in the present work was constructed from the (19 s 14 p 10 d 5 f ) Hg basis set of Gropen^48 in the following
J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 Polarizabilities of RuO 4 , OsO 4 , and HsO 4 1415
way. In the original basis set the most diffuse set of primitive d -type Gaussian type functions ~GTFs! was removed to avoid orthogonality problems. One s -type, four p - and d -type, and three f -type sets of diffuse GTFs were added in a well tempered sequence using an exponent ratio of 2.5. Five s -type GTFs ~#4 to #8 in the original basis! were contracted to two s -type basis functions according to a 3/2 pattern. The eight inner p -type GTFs were contracted to three p -type basis functions using a 3/3/2 contraction pattern. Seven tight d -type sets of GTFs were contracted according to a 3/2/1/ pattern and the four most tight f -type sets of GTFs were contracted to one set. The resulting @ 17 s 14 p 9 d 8 f # basis set is sufficiently flexible to be compared with the results of numeric DF calculations. The data listed in Table I show that in the nonrelativistic HF calculation the exact value^45 of 80 bohr 3 is indeed reproduced. The IORAmm/HF result is in a very good accord with the exact relativistic value^45 and is in fact better than that obtained in the relativistic random phase approximation^46 and DK^47 calculations. The calculations of dipole moment and static electric dipole polarizability of hydrogen halides were carried out with modified basis sets designed by Sadlej^49 for the calcu- lation of molecular electric properties. The original basis sets for elements H(6 s 4 p )@ 3 s 2 p #, F(10 s 6 p 4 d )@ 5 s 3 p 2 d #, Cl(14 s 10 p 4 d )@ 7 s 5 p 2 d #, Br(15 s 12 p 9 d 4 f )@ 9 s 7 p 4 d 2 f #, and I(19 s 15 p 12 d 4 f ) @ 11 s 9 p 6 d 2 f # were used in their un- contracted form. The basis set for At was only partially con- tracted in the following way. Starting from the completely uncontracted basis set, the five innermost s -type primitive GTFs were contracted to one s -type basis function using the coefficients for the first s -AO calculated with the original uncontracted basis set. The same was done for the five inner- most p -type GTFs, the four innermost d -type GTFs, and the two innermost f -type GTFs, which were contracted to one p -, one d - and one f -set of basis functions. Because of orthogo- nality problems, four s -type GTFs ~#13 to #16 in the original set! were replaced with three s -type GTFs with exponents obtained from that of the s -type GTF #12 in a well-tempered sequence with ratio 2.5. The most diffuse s -type GTF and the d -type GTF #11 from the original set were dropped because of orthogonality problems. The resulting @ 14 s 13 p 10 d 4 f # ba-
sis set was used for the At calculations rather than the un- contracted (20 s 17 p 14 d 5 f ) basis set. It should be noted that the contractions of GTFs were only applied in the deep core region so that they do not affect the calculation of dipole moment and polarizability, which depend primarily on the valence and sub-valence orbitals. The results of IORAmm/HF and nonrelativistic HF cal- culations of the dipole moment and the static electric dipole polarizability of hydrogen halides FH through AtH are listed in Table II along with the results of recent time-dependent HF ~TD-HF!, time-dependent Douglas–Kroll ~TD-DK!, and time-dependent Dirac–Hartree–Fock ~TD-DHF! calculations by Norman et al.^15 performed for the same properties. As it has already been noted by Norman et al. ,^15 relativistic effects on the polarizability are extremely weak, just of the order of a few percent. Furthermore, the spin–orbit interaction does not play an important role for the polarizability of closed- shell species^15 and the SR approximation, adopted here and by Norman et al. for the TD-DK method,^15 is sufficient for describing this property correctly. Generally, there is a reasonably good match between the
TABLE I. Static electric dipole polarizabilities a ~bohr^3! of noble gas atoms and the Hg atom. Calculations employ the uncontracted and augmented Ahlrichs TZV basis set ~Ref. 41! unless noted otherwise.
Atom Hartree–Focka^ IORAmm a^ Dirac–Fock b^ Other methodsb
He 1.316 75 1.316 57 1.316 55 1.316 55; c^ 1.31655d Ne 2.3598 2.3625 2.3625 2.3624; c^ 2.3624d Ar 10.707 10.719 10.720 10.718;c^ 10.718d Kr 15.63 15.59 15.64 15.61; c^ 15.61d Xe 26.6 26.3 26.5 26.3; c^ 26.3d Hg 80.0 e^ 43.1e^ 43.0f^ 44.9;g^ 45.26h
a (^) This work. bTaken from Ref. 42. c (^) Direct perturbation theory. Taken from Ref. 42. dScaled ZORA. Taken from Ref. 42. e (^) Calculated with a @ 17 s 14 p 9 d 8 f # basis set ~see text for details!. f (^) Numeric Dirac–Fock value from Ref. 45. gRelativistic RPA value from Ref. 46. hDK-HF value from Ref. 47.
TABLE II. Dipole moments m ~bohr electron! and static electric dipole polarizabilities a ~bohr^3! of the hydrogen halides XH. All calculations em- ploy the uncontracted pVTZ basis of Sadlej. ~Ref. 45! Interatomic distances are taken from Ref. 15.
Method Reference m z a a'b ai^ c a ¯^ d FH Hartree–Fock This work 2 0.7684 4.52 5.87 4. IORAmm/HF This work 2 0.7665 4.52 5.88 4. TD-HF 15 2 0.7682 4.51 5.87 4. TD-DKe^15 2 0.7668 4.52 5.88 4. TD-DHF 15 2 0.7667 4.52 5.88 4. ClH Hartree–Fock This work 2 0.4849 16.20 18.37 16. IORAmm/HF This work 2 0.4783 16.23 18.40 16. TD-HF 15 2 0.4849 16.20 18.37 16. TD-DK e^15 2 0.4799 16.23 18.39 16. TD-DHF 15 2 0.4794 16.23 18.40 16. BrH Hartree–Fock This work 2 0.3788 22.58 25.05 23. IORAmm/HF This work 2 0.3550 22.58 25.06 23. TD-HF 15 2 0.3823 22.57 25.06 23. TD-DK e^15 2 0.3638 22.58 25.07 23. TD-DHF 15 2 0.3610 22.65 25.15 23. IH Hartree–Fock This work 2 0.2563 34.55 37.68 35. IORAmm/HF This work 2 0.2005 34.24 37.40 35. TD-HF 15 2 0.2553 35.59 37.71 35. TD-DK e^15 2 0.2100 34.31 37.45 35. TD-DHF 15 2 0.1971 34.43 37.69 35. AtH Hartree–Fock This work 2 0.1956 41.86 45.14 42. IORAmm/HF This work 2 0.0297 40.78 43.87 41. TD-HF 15 2 0.1944 41.94 45.17 43. TD-DK e^15 2 0.0538 40.77 43.82 41. TD-DHF 15 1 0.0621 42.15 46.00 43. a z -Component of dipole moment. Molecule is oriented along z -axis with the hydrogen atom at the origin. bTransverse component of polarizability. c (^) Longitudinal component of polarizability. dAverage polarizability. e (^) TD-HF results with spin-averaged Douglas–Kroll one-electron Hamil- tonian.
1416 J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 M. Filatov and D. Cremer
cients of the 2 p -, 2 p , 3 p -, 4 p -, and the 5 p -orbital obtained with the original set. Fourteen d -type GTFs were contracted to five d -functions ~contraction pattern 2/3/3/3/3! utilizing the coefficients taken from the 3 d -, 3 d -, 3 d -, 4 d -, and 5 d -orbital of the original set. Twelve f -type GTFs were con- tracted to four f -functions ~contraction pattern 4/3/3/2! em- ploying the coefficients of the 4 f -, 4 f -, 5 f -, and 5 f -orbital of the original set. Although the basis set used for hassium was not specifically optimized for this element, the contrac- tion scheme applied results in a basis set, which is suffi- ciently flexible in the valence and sub-valence regions to guarantee the calculation of reliable electric response prop- erties.
IV. RESULTS AND DISCUSSION
Static electric dipole polarizabilities of group VIII metal tetroxides calculated with the quasirelativistic IORAmm as well as the nonrelativistic HF and MP2 methods are listed in Table III. The only experimentally known value is the polar- izability of OsO 4 of 55.13 bohr^3.^57 The IORAmm/MP2 value of 56.57 bohr^3 is in a reasonable agreement with experiment, whereas HF ~both quasirelativistic and nonrelativistic! under- estimates substantially the polarizability. However, all meth- ods employed ~correlated, uncorrelated, quasirelativistic, nonrelativistic! suggest an ordering of the polarizabilities of group VIII metal tetroxides according to RuO 4 .OsO 4 .HsO 4. The origin of such an ordering can be traced back to variations of the dipole electronic transition energies from the ground 1 A 1 state to excited 1 T 2 states, which are pre- sented in Table IV. These excitation energies play an impor- tant role for the static electric dipole polarizability as re- flected by Eq. ~ 27!^2
a ¯^ (^5) ( k fi 0
f (^) 0 k ~ D E 0 k!^2
where f (^) 0 k is the dimensionless oscillator strength and D E 0 k the corresponding excitation energy ~given in hartrees!. Al- though the CIS method,^53 which is employed to study the excitation energies, does not include dynamic electron corre- lation corrections and, therefore, cannot be expected to yield accurate excitation energies, it provides the qualitative trends of the excitation energies reasonably well ~see Table IV!. The data in Table IV show that the lowest 1 T 2 ←^1 A 1 excitation energy increases in the sequence RuO 4 ,OsO 4 ,HsO 4. At the nonrelativistic level, this increase is associated with in-
creasing ionicity of M–O bonds in the tetroxides ~see NBO metal charges in Table III!. The highest occupied molecular orbitals of MO 4 are the ligand-centered t 1 - and t 2 -symmetric orbitals, whereas the metal d -orbitals ~ e - and t 2 -symmetric! are empty. As the ligand charge becomes more negative in the sequence RuO 4 ,OsO 4 ,HsO 4 ~see Table III!, the empty metal d -orbitals are increasingly destabilized in the electric field of ligands. Relativity adds to this trend destabilizing the metal d-orbitals further.8 –10^ This is reflected in the 1 T 2 ←^1 A 1 excitation energies presented in Table IV. The relativ- istic blue shift ~with respect to the nonrelativistic values! of the excitation energies is especially pronounced in HsO 4 where it reaches ;2 eV for the lowest dipole transitions. Hence, the dipole polarizabilities of MO 4 calculated with IORAmm are smaller than the nonrelativistic polarizabilities. The quasirelativistic calculations in the present work are performed at the SR level, neglecting all spin-dependent relativistic effects of which the most important are spin– orbit ~SO! interactions. The inclusion of SO interactions re- sults in splitting of the atomic d -levels into d 5/2- and d 3/2-sublevels. However, the magnitude of this splitting is smaller than the SR destabilization of d -orbitals with respect to the nonrelativistic values. For example, the four- component DHF calculations of Malli^32 yielded 4.35 eV for the SR destabilization of 6 d -orbitals of Hs, whereas the SO splitting between 6d5/2- and 6 d 3/2-orbitals was only 1.74 eV ~2.22 and 1.05 eV for SR-destabilization and SO-splitting of Os, respectively!.^32 The SO interaction, which is missing in the IORAmm calculations, could have resulted in the split- ting of dipole excitation energies around the values reported in Table IV into lines shifted upward and downward in en- ergy, such that the overall effect of the SO interactions on the polarizabilities should be minimal and should not overturn the trend obtained in the SR IORAmm calculations. In general, the static electric dipole polarizabilities ob- tained in the IORAmm calculations appear to be more accu- rate than the theoretical values obtained previously with the help of relativistic DFT calculations.^30 Despite the use of the full four-component formalism, the values of 43.73 (RuO 4 ), 40.22 (OsO 4 ), and 42.24 bohr^3 (HsO 4 ) obtained by Pershina et al.^30 are far too low compared to the experimental data available.^57 Apart from a large underestimation of the polar- izability of OsO 4 , the relativistic DFT calculations predict a trend RuO 4 .OsO 4 ,HsO 4 , which differs from that obtained in the present work. Given that conventional density func-
TABLE IV. Low-energy electronic dipole transition energies ~in eV! and dimensionless oscillator strengths in tetroxides of group VIII elements Ru, Os, and Hs ~Z 5108 !.
Molecule Method A 21 T 2 ← X 21 A 1 B 21 T 2 ← X 21 A 1 C 21 T 2 ← X 21 A 1 RuO 4 a^ CIS 4.05/0.0079 5.37/0.0215 5.84/0. IORAmm/CIS 4.28/0.0102 5.71/0.0274 6.05/0. OsO 4 b^ CIS 4.66/0.0043 6.06/0.0334 6.97/0. IORAmm/CIS 5.38/0.0116 6.96/0.0504 7.48/0. HsO 4 c^ CIS 6.06/0.0004 7.64/0.0534 9.04/0. IORAmm/CIS 8.14/0.0193 9.68/0.0832 10.31/0. aCalculated at the experimental M–O bond length of 1.706 Å. bCalculated at the experimental M–O bond length of 1.714 Å. cCalculated at the IORAmm/MP2 bond length of 1.726 Å.
1418 J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 M. Filatov and D. Cremer
tionals have not an exceptionally good record when calculat- ing electric response properties,^28 the results of Pershina et al.^30 can not be considered reliable. Yet another trend in polarizabilities of MO 4 ~M 5 Ru, Os, Hs! was obtained by Du¨llmann et al. ,^31 who used for OsO 4 the experimental value of 55.13 bohr^3 , for HsO 4 the value ~57.15 bohr^3! from relativistic DFT calculations of Pershina et al.^30 corrected for the difference between the calculated and experimental polarizabilities of OsO 4 , and for RuO 4 a value ~53.45 6 1.01 bohr^3! obtained by extrapolation from po- larizability of OsO 4 assuming that the polarizability is pro- portional to the molar volume. Thus, a trend RuO 4 ,OsO 4 ,HsO 4 precisely opposite to that of the present work was obtained. However, such a trend, which is based on both the results of unreliable calculations and a questionable extrapo- lation procedure^58 rather than direct measurements, can hardly be considered as being reliable. The static electric dipole polarizabilities obtained by Per- shina et al.^30 and Du¨llmann et al.^31 were used to estimate the adsorption enthalpies of metal tetroxides on the quartz sur- face under the assumption that only physical adsorption ~due to dispersion forces! takes place^59 and that the MO 4 mol- ecules behave like spherically symmetric atoms. Because the experimental adsorption enthalpies vary in the order RuO 4 ,OsO 4 ,HsO 433 and because the energy of physical adsorp- tion of a spherically symmetric atom is roughly proportional to electric dipole polarizability,^59 the same ordering of the polarizabilities of MO 4 was advocated by Du¨llmann et al.^31 and Pershina et al.^30 However, the results of the present work suggest that the trend in the measured adsorption enthalpies has to be rationalized on the basis of a more detailed model for adsorption, which takes into account molecular structure rather than just the polarizabilities and dispersion forces of the MO 4 molecules. In particular, the bond ionicity should be considered because it relates with electrostatic interactions possible between MO 4 and quartz surface. It should be noted that the metal charge in MO 4 correlates nicely with the ad- sorption enthalpies as has been established in earlier work by Pershina^29 and by Malli,^32 respectively. In the present inves- tigation, the NBO charges on metal obtained with the help of IORAmm/MP2 calculations ~see Table III! are in line with the trend in the observed adsorption enthalpies.
IV. CONCLUSIONS
Within the regular approximation to the full four- component relativistic Hamiltonian, the analytic expressions for the derivatives of the total molecular energy with respect to an external electric field are derived and presented in ma- trix form suitable for implementation in standard quantum- chemical codes. The formalism developed is tested in ab initio calculations of electric dipole moments and static elec- tric dipole polarizabilities for a number of compounds con- taining heavy elements. The benchmark calculations demon- strate that the formalism presented enables one to obtain for compounds containing heavy elements results, which closely match the exact four-component relativistic results at essen- tially the same price as that of conventional nonrelativistic calculations. It is gratifying that with the use of the
IORAmm method both very weak and very strong effects of relativity on atomic and molecular polarizabilities can be successfully described. The formalism developed was applied to study the static electric dipole polarizabilities of group VIII metal tetroxides MO 4 for M 5 Ru, Os, Hs ~Z 5108 !. The polarizabilities ob- tained at the IORAmm/MP2 level of theory decrease in the sequence RuO 4 .OsO 4 .HsO 4 , which is different from those obtained in earlier studies based on relativistic DFT calculations.30,31^ On the basis of the trend obtained in the present work, it is suggested to reconsider dispersion models of adsorption of MO 4 on a quartz surface in favor of more elaborate models which consider both electrostatic and dis- persion interactions. For example, the adsorption enthalpy correlates well with the M–O bond ionicity suggesting an electrostatic component of adsorption.
ACKNOWLEDGMENTS
This work was supported by the Swedish Research Council ~Vetenskapsra˚det!. An allotment of computer time at the National Supercomputer Center ~NSC! at Linko¨ping is gratefully acknowledged.
(^1) C. J. E. Boettcher, Theory of Electric Polarization ~Elsevier, Amsterdam, 1973 !. (^2) T. M. Miller and B. Bederson, Adv. At. Mol. Phys. 13 , 1 ~ 1977 !. (^3) A. H. de Vries, P. Th. van Duijnen, R. W. Zijlstra, and M. Swart, J. Electron Spectrosc. Relat. Phenom. 86 , 49 ~ 1997 !. (^4) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular vibrations: The Theory of Infrared and Raman Vibrational spectra ~Dover, New York, 1980 !. (^5) S. W. Rick and S. J. Stuart, Rev. Comput. Chem. 18 , 89 ~ 2002 !. (^6) U. Norinder and T. O¨ sterberg, Perspect. Drug Discovery Des. 19 , 1 ~ 2000 !. (^7) M. Broyer, R. Antonine, E. Benichou, I. Compagnon, P. Dugourd, and D. Rayane, Comp. Rend. Phys. 3 , 301 ~ 2002 !. (^8) P. Pyykko¨, Chem. Rev. 88 , 563 ~ 1988 !; 97 , 597 ~ 1997 !. (^9) W. H. E. Schwarz, in Theoretical Models of Chemical Bonding. Part 2. The concept of the Chemical Bond , edited by Z. B. Maksic´^ ~Springer- Verlag, Berlin, 1990!, p. 593. (^10) J. Almlo¨f and O. Gropen, in Reviews in Computational Chemistry , edited by K. B. Lipkowitz and D. B. Boyd ~VCH, New York, 1996!, Vol. 8, p.
(^11) V. Kello¨, A. J. Sadlej, and B. A. Hess, J. Chem. Phys. 105 , 1995 ~ 1996 !. (^12) L. Visscher, T. Saue, and J. Oddershede, Chem. Phys. Lett. 274 , 181 ~ 1997 !. (^13) A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, Phys. Rev. Lett. 82 , 3589 ~ 1999 !. (^14) K. G. Dyall, Int. J. Quantum Chem. 78 , 412 ~ 2000 !. (^15) P. Norman, B. Schimmelpfennig, K. Ruud, H. J. Aa. Jensen, and H. A˚ gren, J. Chem. Phys. 116 , 6914 ~ 2002 !. (^16) M. Pecul and A. Rizzo, Chem. Phys. Lett. 370 , 578 ~ 2003 !. (^17) P. A. M. Dirac, Proc. R. Soc. London, Ser. A 117 , 610 ~ 1928 !. (^18) M. Douglas and N. M. Kroll, Ann. Phys. ~N.Y.! 82 , 89 ~ 1974 !; B. A. Hess, Phys. Rev. A 32 , 756 ~ 1985 !; 33 , 3742 ~ 1986 !. (^19) Ch. Chang, M. Pe´lissier, and P. Durand, Phys. Scr. 34 , 394 ~ 1986 !; J.-L. Heully, I. Lindgren, E. Lindroth, S. Lundqvist, and A.-M. Ma˚rtensson- Pendrill, J. Phys. B 19 , 2799 ~ 1986 !; E. van Lenthe, E. J. Baerends, and J. G. Snijders, J. Chem. Phys. 101 , 9783 ~ 1994 !. (^20) K. G. Dyall and E. van Lenthe, J. Chem. Phys. 111 , 1366 ~ 1999 !. (^21) M. Filatov, Chem. Phys. Lett. 365 , 222 ~ 2002 !. (^22) M. Filatov and D. Cremer, J. Chem. Phys. 118 , 6741 ~ 2003 !. (^23) A. Wolf, M. Reiher, and B. A. Hess, J. Chem. Phys. 117 , 9215 ~ 2002 !. (^24) M. Filatov and D. Cremer, Phys. Chem. Chem. Phys. 5 , 1103 ~ 2003 !. (^25) M. Filatov, Chem. Phys. Lett. 373 , 131 ~ 2003 !. (^26) See, e.g., G. T. Velde, F. M. Bickelhaupt, E. J. Baerends, C. F. Guerra, S.
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