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Electric field dependence of magnetic properties: Multiconfigurational self-consistent field calculations of hypermagnetizabilities and nuclear shielding polarizabilities of N2 , C2H2 , HCN, and H2O Antonio Rizzo Istituto di Chimica Quantistica ed Energetica Molecolare del Consiglio Nazionale delle Ricerche, Via Risorgimento 35, I-56126 Pisa, Italy Trygve Helgaker and Kenneth Ruud Department of Chemistry, University of Oslo, P.O.B. 1033 Blindern, N-0315, Oslo, Norway Andrzej Barszczewicz and Michal” Jaszuński Institute of Organic Chemistry, Polish Academy of Sciences, 01 224, Warszawa, ul. Kasprzaka 44, Poland Poul Jo” rgensen Department of Chemistry, Aarhus University, DK-8000, Aarhus C, Denmark ~Received 23 December 1994; accepted 24 February 1995! Multiconfigurational self-consistent field ~MCSCF! response is used to study the electric field dependence of magnetizabilities and nuclear shielding constants for N2 , C2H2 , HCN, and H2O. London perturbation-dependent atomic orbitals are used to ensure gauge origin independence. The computed magnetizabilities and shielding derivatives show a strong electron correlation dependence. The N2 results confirm the conclusions of previous ab initio studies. For the other molecules, this is the first study of the above magnetic properties beyond the SCF approximation. © 1995 American Institute of Physics. I. INTRODUCTION Recent developments in theoretical and computational methods have enabled ab initio studies of a wide variety of molecular electric, magnetic, and optical properties. In particular, properties arising from the nonlinear response of the molecule to a combination of electric and magnetic fields can now be calculated. The theory needed to describe these properties has been known for many years ~see, for example, the reviews by Buckingham,1 Buckingham and Orr,2 Bishop,3 and Raynes4!. In practice, however, the calculations have been hampered by a strong dependence on such aspects as the choice of the gauge origin, the choice of the basis set, and the description of electron correlation. In this work we study the electric field dependence of magnetic properties. The problems encountered in the calculation of magnetic properties, such as magnetizabilities and NMR shielding constants, are reflected in the unphysical dependence of the calculated values on the chosen gauge origin. Even at the SCF level, it is practically impossible to ensure gauge origin independence of the results for a polyatomic molecule, unless methods specially formulated for this type of calculations are used. In this work, we apply atomic basis sets that are explicitly dependent on the external magnetic field, the so-called London atomic orbitals5 ~LAOs, or GIAOs, gauge invariant atomic orbitals6!. Efficient methods that enable practical use of these orbitals have only recently been formulated, first for SCF functions7 and more recently for arbitrary approximate wave function.8 The theory of linear response for multiconfigurational selfconsistent field ~MCSCF! wave functions9 has been successfully combined with the use of LAOs.10–13 Nuclear shieldings using London orbitals have also been calculated at second, third, and partly fourth14 –16 order in perturbation J. Chem. Phys. 102 (22), 8 June 1995 theory and using coupled cluster singles and doubles wave functions.17 There is one aspect of the theory we shall discuss in some detail in this work. For electric field derivatives it is necessary to use a new formulation of the approach,18,19 applying what is called the natural connection for perturbationdependent basis sets. This is needed in order to avoid numerical problems arising when large basis sets are used and, at the same time, high accuracy of the computed magnetic properties is required. In this work the electric field dependence of the magnetic properties is analyzed performing finite-field calculations. Numerical differentiation of the magnetizabilities and shielding constants with respect to the strength of the applied electric field requires high accuracy. The final third and fourth order properties are thus obtained in a mixed analytical-numerical scheme. The experimental quantity that may give an estimate of the magnetizability polarizability ~hypermagnetizability, h! is the Cotton–Mouton constant, related to the Cotton– Mouton effect20—the birefringence of light in gases in a constant magnetic field. Isotropic substances show weak birefringence when radiation passes through a sample in a direction orthogonal to a strong magnetic field. This effect, which is analogous to the electric Kerr effect, is commonly expressed by experimentalists through the relation21 Dn5n i 2n' 5C CMlB 2 linking the observed birefringence ~anisotropy of the refraction index n! to the magnetic field flux B and the wave length l through the so-called Cotton–Mouton constant C CM . Experimentally, one measures the ellipticity gained by polarized light passing through the sample in a very strong 0021-9606/95/102(22)/8953/14/$6.00 © 1995 American Institute of Physics 8953 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8954 and uniform magnetic field. The resulting magneto-optic effect is often extremely weak, making its accurate observation very difficult. The shielding polarizabilities provide an approximate description of the effects of inter- and intramolecular electric fields on the shielding. They can be used to estimate the role of solvent effects ~see, for example, Ref. 22! or the interaction with a distant part of the molecule ~see, e.g., Ref. 23!. We refer to some recent reviews4,24,25 for a more detailed discussion of the calculations of shielding polarizabilities, their applications and a bibliography of numerous earlier works. We first briefly review the definitions of the studied properties. Next, we present the formulas used to compute these properties within the LAO approach. We then present some computational details, and finally analyze the results for the electric field derivatives of the magnetizabilities and shielding constants. II. THEORY A. Definition of magnetizability and shielding polarizabilities The quantities whose dependence on the electric field is studied in this paper are the molecular magnetizability26 x and the nuclear magnetic shielding s (K), defined as x 52 ] 2 e ~ B,m! ] 2B s ~ K ! 511 U ~1! , B50 ] 2 e ~ B,m! ] B ] mK U ~2! , B5mK 50 where e~B,m! is the molecular energy in the presence of the external magnetic field induction B and the nuclear magnetic moments m. mK is the nuclear magnetic moment of the Kth nucleus. Switching on the electric field perturbation E, the elements of the molecular magnetizability tensor xab may be written as E x ab 5 x ab 1 j ab , g E g 1 21 h gd , ab E g E d 1••• , ~3! where the hypermagnetizability tensors jab,g and hab,gd are given as j ab , g 5 E ]x ab , ]Eg h ab , gd 5 B E ] 2 a ab ] 2 x gd 5 . ] B g] B d ] E a] E b ~4! E 8 ,g~ K ! E g s ab ~ K ! 5 s ab ~ K ! 1 s ab 1 1 2 9 , gd ~ K ! E g E d 1••• , s ab where the so-called shielding polarizabilities E ]s ab ~K! 8 ,g~ K ! 5 s ab ]Eg In the first definition of hab,gd the electric dipole polarizability tensor aab is introduced. The second definition, taken from Eq. ~3! above, is directly related to the technique used in this work to compute hab,gd , in the sense that hab,gd is obtained by numerical differentiation of analytically calculated magnetizabilities. Similarly, the shielding tensor for nucleus K may be expanded as ~7! and 9 , gd ~ K ! 5 s ab E ] 2 s ab ~K! ] E g] E d ~8! describe the effect of the electric field on the magnetic shielding tensor. In the following the index K will be dropped when no confusion can arise. The theory for the calculation of magnetizabilities and nuclear magnetic shielding constants has been thoroughly described in previous papers ~see, e.g., Refs. 12 and 13!. Here, we compute electric field derivatives of these observables. The theory has some new aspects as discussed in the following. B. Calculation of magnetizability and shielding polarizabilities The Hamiltonian for a molecule in the presence of an external magnetic field and nuclear magnetic moments can be written as ~atomic units! H5 1 2 ( p 2i 2 ( i i,K ZK 1 1 r i,K 2 ( iÞ j 1 , rij ~9! where the kinetic momentum of the ith electron is given as p i 52i“ i 1A~ ri ! ~10! and the dependence on the magnetic fields is collected in the vector potential A~ r! 5 1 B3rO 1 a 2 2 ( K mK 3rK r 3K , ~11! where a is the fine structure constant and O denotes the ~arbitrary! gauge origin. The summation runs over all the atoms in the molecule. If a perturbation is applied to a molecular system, the system responds. It is advantageous to try to mimic this physical response in our basis functions. For external magnetic fields this can be achieved by introducing the London atomic orbitals5 e ~5! ~6! v m ~ rM ;AM ! 5e ~ 2iAM –r! x m ~ rM ! , ~12! AeM where represents the potential at the position of the nucleus M AeM 5 12 B3RM O ~13! and xm is a conventional Gaussian basis function positioned on that nucleus. The London orbitals respond correctly to the external magnetic field in the sense that they are correct through first order in the external magnetic field for a oneelectron, one-center system.11 J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties As shown by Helgaker and Jo” rgensen,8 the Hamiltonian integrals that occur for London atomic orbitals can be written in the following form: 1 2 ^ v m ~ AeM ! u p 2 2 ( K e 5 ^ x m u e iAM N –r S ZK u v n ~ AeN ! & rK 1 @ 2i“1A~ r! 2AeN # 2 2 2 ( K D ZK u x n& , rK ~14! g mnrs ~ A! 5 ^ x m x n u exp~ iAeM N –r1 1iAeRS –r2 ! r 12 u x rx s& , ~15! where the new vector potentials are defined as AeM N 5AeM 2AeN 5 21 B3BM N , 1 A~ r! 2AeN 5 B3rN 1 a 2 2 ( K ~16! mK 3rK r 3K . H int52E–de , ~17! ~18! where de is defined as ( ri . ~19! i As the operator in Eq. ~18! commutes with the exponential operator in the London atomic orbital, the one-electron integrals we need can be written as e h mn ~ A! 5 ^ x m u e ~ iAM N –r! 2 ( K S 1 @ 2i“1A~ r! 2AeN # 2 2 D ZK 2E–de u x n & , rK We now follow closely the exposition in a recent paper on MCSCF magnetizabilities using London atomic orbitals.27 Only the major steps are given, with emphasis on the differences arising when the electric field effects are investigated. The wave function perturbed by the external magnetic field and the nuclear magnetic moments can be expressed in terms of a unitary rotation of orbital and configuration parameters in the reference wave function optimized at zero field and zero nuclear magnetic moments u WF~ B,m! & 5exp~ i k ! exp~ iS ! u RWF& . ~21! Our reference wave function is optimized at a given electric field E and constructed as a linear combination of Slater determinants u RWF& 5 ( C bu b & . ~22! b We note that there is no dependence on the gauge origin in these integrals. Whereas this Hamiltonian is adequate for the nuclear shielding constants and magnetizabilities, an additional term must be introduced in the one-electron integrals when we want to study the electric field derivatives of these properties by a finite field approach. The expressions for the magnetizabilities and nuclear shieldings are obtained by explicit ~analytic! differentiation of our modified Hamiltonian with respect to nuclear magnetic moments and the magnetic field. The third and fourth derivatives are then determined by finite difference of the results obtained at different electric field strengths. As we shall see, the use of London atomic orbitals ensures gauge origin independent third- and fourth-order mixed electric and magnetic field derivatives. The effect of an external electric field can be represented by the interaction between the molecular dipole operator de and the external electric field as de 52 8955 The orbital and configuration operators k and S in Eq. ~21! describe magnetic perturbations and are therefore given by k 5 ( k Rrs ~ E rs 1E sr ! , ~23! r.s S5 ( S RK ~ u K &^ RWFu1uRWF&^ K u ! . ~24! KÞ0 Only nonredundant rotations are included in the summation of Eq. ~23!. The one-electron excitation operators in Eq. ~23! are defined as ( E mn 5 s 561/2 1 am sa ns . ~25! Our reference wave function is optimized at zero magnetic field and nuclear magnetic moments, but at a finite electric field. In this case the Hamiltonian integrals in Eqs. ~15! and ~20! reduce to the ordinary field-free integrals with an extra interaction operator describing the finite electric field. Thus our reference function has an implicit dependence on the external electric field. The Slater determinants in Eq. ~22! are written as ordered products of creation operators ub&5 ) a b1 u vac& i ~26! i creating electrons in a space of orthonormalized molecular orbitals ~OMOs! f OMO ~ B! 5 ( f UMO ~ B! T nm ~ B! . m n ~27! n ~20! where we have added the electric field interaction integral to Eq. ~14!. No modification is needed in the expression for the two-electron integrals. The unmodified molecular orbitals ~UMOs! f UMO (B) corren spond to the optimized orbitals at B50 and m50 and are linear combinations of London atomic orbitals f UMO ~ B! 5 ( C n m v m ~ B! . n ~28! m J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8956 We note that there is no field dependence in the MO coefficients. All dependence on the external magnetic field in our wave function is collected in the London orbitals and the connection matrix T mn . As discussed by Olsen et al.,18 it is advantageous to choose this connection as T5W 21 ~ WS 21 W 1 ! 1/2 , ~29! where S is the UMOs overlap matrix S mn ~ B! 5 ( C m m C n n ^ v m ~ B! u v n ~ B! & ~30! mn ( C n n C m m ^ x n u v m ~ B! & . ~31! mn This is the so-called natural connection which ensures that the OMOs change as little as possible when the magnetic field is turned on. This is what we expect our orbitals to do, and it means that the natural connection properly describes the physical response of the London orbitals to the external magnetic field. The use of the natural connection leads to a numerically stable algorithm, in contrast to the symmetric connection.18,19 It is not obvious from Eqs. ~29! to ~31! that the natural connection gives gauge origin independent results, as it involves the use of integrals W mn ~B! that depend on the gauge origin. This gives an origin dependence in our Hamiltonian that parallels that of the exact Hamiltonian. And, as shown by Olsen et al.,18 all observable properties calculated from the Hamiltonian in the natural connection are strictly gauge origin independent. Our first attempt at calculating shielding polarizabilities and hypermagnetizabilities was by the use of the symmetric connection. However, as this approach is numerically unstable,18,19 it turned out to be impossible to perform the numerical differentiation from the shielding constants and magnetizabilities evaluated at the different electric field strengths. Thus the evaluation of higher order properties like shielding polarizabilities and hypermagnetizabilities provides a striking example of the need for a stable algorithm for the calculation of second-order properties. Combining the molecular orbitals in Eq. ~27!, with the Hamiltonian integrals in Eqs. ~15! and ~20!, our Hamiltonian, in accordance with Helgaker and Jo” rgensen,8 can be written as H ~ B! 5 1 ( h̃ mn~ B! E mn 1 2 ( mn g̃ mnpq ~ B! e mn pq , ~32! mnpq where the molecular integrals and two-electron excitation operator are defined as h̃ mn ~ B! 5 ( m8n8 * ~ B! h m 8 n 8 ~ B! T n 8 n ~ B! , T mm 8 m8n8r8s8 * ~ B! T *nn ~ B! g m 8 n 8 r 8 s 8 ~ B! T mm 8 8 3T r 8 r ~ B! T s 8 s ~ B! , e mn pq 5E mn E pq 2 d pn E mq . ~34! ~35! As described in Refs. 13 and 27, the nuclear shieldings and magnetizabilities in an external electric field can now be calculated by straightforward analytic differentiation of the energy functional 3exp~ i k ! u RWF& . ( C n n C m m ^ v n~ B0 ! u v m~ B! & mn 5 ( e ~ B,m! 5 ^ RWFu exp~ 2i k ! exp~ 2iS ! H exp~ iS ! and the nonsymmetric matrix W is given by W mn ~ B! 5 g̃ mnrs ~ B! 5 ~33! ~36! The nuclear shieldings are obtained by differentiating once with respect to nuclear magnetic fields and once with respect to external magnetic field, whereas the magnetizability is obtained by differentiating twice with respect to the external magnetic field. In calculations without London orbitals this would result in the ordinary perturbation expression with MO coefficients modified due to the presence of the electric field. In calculations using London orbitals we obtain, in addition to the modified integrals presented elsewhere,12,27 an extra derivative contribution from the electric field interaction operator. Thus the first and second derivatives of the one-electron Hamiltonian look like U ] h mn ]B 1 5 ^ x m u LN u x n & 1iQ M N ^ x m u rh u x n & 2 B50 2iEQ M N ^ x m u rdu x n & , U ] 2 h mn ] B2 ~37! 5 ^ x m u r 2N I2rN rTN u x n & 1iQ M N ^ x m u rLTN u x n & B50 1Q M N ^ x m u rrT h u x n & Q M N 2EQ M N ^ x m u rrT du x n & Q M N , ~38! where h represents the ordinary one-electron Hamiltonian without contributions from external magnetic and electric fields, the overbar means that the integral is to be symmetrized, LN is the operator for angular momentum around nucleus N, LN 52irN 3“, and the elements of the matrix Q M N are the components of the vector from nucleus N to nucleus M , see Ref. 8. The last terms in Eqs. ~37! and ~38! are new to our implementation, and vanish in the absence of an external electric field. We notice that both derivative integrals are independent of the gauge origin, and that there is no change in the twoelectron integrals, as the electric field interaction operator is a one-electron operator. The new integrals that appear due to the presence of the electric field can be quite easily evaluated following the outline in Ref. 12. Thus a numerically stable algorithm for gauge origin independent magnetizabilities and shielding constants has been presented. The shielding polarizabilities and hypermagnetizabilities can be obtained by calculations of the nuclear J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8957 shieldings and magnetizabilities at different electric field strengths, followed by differentiation with respect to the electric field strength. C2 m2 J21 ~SI, MKSA!51.481 847310225 ~4pe0! cm3 ~emu!; —1 a.u. of x 5(e 2 a 20 )/(m e )57.891 04310229 J T22 or C. Relationship to experiment 7.891 04310230 cm3; —1 a.u. of j 5(e 3 a 30 )/(m e E h )51.534 562310240 Before concluding this section, we give some definitions needed to relate the quantities we compute to experiment, in particular the hypermagnetizability h to the Cotton–Mouton effect. The first theoretical treatment of the Cotton–Mouton effect appeared in 1910.28 A fundamental contribution to the understanding of the topic was given by Buckingham and Pople in 1956.29 A classical statistical mechanics approach30 shows that for rigid diamagnetic molecules the so-called Cotton–Mouton constant m C, proportional to the mean electric polarizability in a strong magnetic field, can be written as m C5 2pN 27 1 H S S 1 1 h ab , ab 2 h aa , bb 5 3 1 1 a ab x ab 2 a aa x bb 5kT 3 D DJ , ~39! where N is Avogadro’s number, k the Boltzmann constant, and T the temperature. The Einstein summation ~here and below! is assumed. For axial molecules the above equation simplifies to m C5 F G 2pN 2 DaDx , Dh1 27 15kT 1 3 C2 m2 J21 T2252.682 108310244 ~4pe0! cm3 G22; —1 a.u. of s 8 5 p pm(a 20 /e)51.944 69310218 mV2155.830 03310214 cm statV21 ~esu!; —1 a.u. of s 9 5 p pm(a 20 /e) 2 53.781 82310230 m2 V2253.398 92310221 cm2 statV22. In Eq. ~40! above the factor relating m C to the quantity in square brackets ~computed in a.u.! is 3.758 738 10221 cm3 G22 mol21 ~4pe0! ~CGSM! or 5.935 561 10231 m5 A22 mol21 ~SI!. The relationship between the experimental Cotton– Mouton constant21 C CM and the m C introduced by Buckingham and Pople,29 at a temperature T, 1 atm and in the CGSM unit system, is C CM~cm21 G22 )5 27 1 • C ~ cm3 G22 mol21 ! 2V m l ~ cm! m 5 0.164 518 C ~ cm3 G22 mol21 ! , T ~ K!•l~cm! m ~40! where Da5ai2a' , Dx5xi2x' , and the hypermagnetizability anisotropy is defined as D h 5 51 ~ h ab , ab 2 C m T2251.379 196310226 ~4pe0! cm4 s21 G21; —1 a.u. of h 5(e 4 a 40 )/(m e E 2h )52.984 25310252 h aa , bb ! ~41! thus differing by a factor of five from that used by Hüttner and collaborators.31 Notice also that some authors ~e.g., Ref. 31! employ König’s32 definition of m C, a factor of 9 larger than Eq. ~39! above. Our results for m C and Dh are given according to Eqs. ~39! to ~41! above. The experimentally measured frequency dependent hypermagnetizability can be written in terms of a combination of quadratic ~diamagnetic contribution! and cubic ~paramagnetic contribution! terms ~r position, L angular momentum, and Q quadrupole moment operators!33 p d h ab , gd ~ 2 v ; v ,0,0 ! 5 h ab , gd 1 h ab , gd 52 1 2 ^^ r a ;r b ,L g ,L d && 2 v ; v ,0,0 2 1 4 ^^ r a ;r b ,Q gd && 2 v ; v ,0 . ~42! By taking advantage of symmetry and thus reducing the expression for the paramagnetic contribution to a Cauchy moment expansion, we were able to study the frequency dependence of the hypermagnetizability anisotropy in the Ne and Ar atoms using the multiconfigurational response method.33,34 The same approach cannot straightforwardly be employed for molecules. Atomic units are used throughout the paper ~unless otherwise stated!. Conversion factors to some of the units used by other authors are given in the following: —1 a.u. of a 5(e 2 a 20 )/(E h )51.648 78310241 where l is the radiation wavelength and V m is the molar volume at the given temperature of the ~ideal! gas. III. COMPUTATIONAL DETAILS A. Computational aspects There are many practical problems to deal with in the calculations, in particular the selection of the finite fields, the choice of basis set and active space for CASSCF and the dependence of the results on the molecular geometry. The accuracy of the numerical differentiation depends crucially on the strength of the applied finite field. Often the field suitable for shielding derivatives ~where first derivatives may be nonzero! is too small to compute reliably the magnetizability derivatives ~where the first derivative may vanish because of higher symmetry!. In such cases, different fields were applied for different properties. Usually field strengths of 0.01 a.u. were suitable for the magnetizability derivatives and 0.001 a.u. for the shielding derivatives. For many tensor components we found intermediate fields suitable for both calculations. We tried to minimize the number of finite fields, since the calculations are time consuming. For example, for s~H! in H2O at least eight separate combinations are required to obtain all tensor components. A complete calculation for each of these combinations took a few CPU hours on a Convex 3840 computer. The convergence of the properties with respect to extension of the basis set and increase of the active space is not systematic. For the individual components the convergence differs, some of them being apparently much easier to compute than others. We shall, therefore, describe in the text the J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 8958 Rizzo et al.: Electric field dependence of magnetic properties main conclusions of this computational analysis for all four molecules. However, as a rule we shall present only the numbers for the best SCF and best correlated calculation. To illustrate the basis set dependence we quote two sets of values for N2 . The correlation and geometry dependence will be discussed for H2O. These are typical examples, and at the same time there are some literature data which we use for comparison. All calculations have been performed using a suite of programs which includes HERMIT35 for the integrals, SIRIUS36 for the wave function and ABACUS37 for the magnetic properties. It is not reasonable to describe all the aspects of the calculation in detail. To begin with, there are too many tensor components for a detailed analysis. An extreme case is the s9 tensor for the H atom in H2O, which has 41 ~28 different! nonvanishing components.38 Following Grayson and Raynes,39,40 we shall discuss for HCN and H2O only those components that do not vanish after rotational averaging. For N2 and C2H2 we include all nonzero first derivatives. B. Basis sets The dependence of the properties on the basis set was studied for each molecule. We began the construction of the basis set with the GTO sets referred to as H IV in Ref. 12. They consist of [11s7 p3d1 f ]/ ^ 8s7p3d1 f & contractions for C, N, and O atoms and a [6s3p1d]/ ^ 5s3p1d & contraction for the hydrogen atom. These sets are well suited for the calculation of magnetizabilities and shielding constants, but they do not include diffuse functions needed for the electricfield derivatives of these properties. In the larger basis sets we add diffuse s, p, d, and f ~s, p, and d for H! functions using a geometric progression for the exponents. As in the calculation of Verdet constants,41 we observe that the first set of diffuse functions, giving rise to the sets labeled IVa here as in Ref. 41, changes the computed properties significantly. The changes upon adding a second set are smaller. These sets, called IVb,41 finally include [13s9 p5d2 f ]/ ^ 10s9p5d2 f & functions for C, N, and O and [8s5p2d]/ ^ 7s5p2d & functions for H. The second f function on C, N, or O and d function on H were included already in set IVa for some molecules and only in set IVb for the others. The basis set convergence was studied for many properties not only at the SCF level, but also at the CASSCF level. We shall illustrate the basis set dependence for N2 . For C2H2 and HCN we were unable to use basis IVb at the correlated level. For C2H2 the differences between the SCF h components computed using basis sets IVa and IVb are smaller than 5% ~we exclude in these comparisons the very small tensor components!, for s8 they do not exceed 1%, and for s9 10%. Also for HCN the differences between the SCF IVa and IVb results are smaller than for N2 . For water the effect of the basis set is larger: about 10% for h and s9 at the SCF level, and significantly larger for s9 at the CAS A ~see below! level ~on the average, about 30%!. To establish convergence with the basis set at the CASSCF level we would therefore need to use a basis set larger than IVb, which is not presently possible. C. Geometries We have used the following geometries: —for N2 , R~N–N!51.097 513 Å;42 —for C2H2 , R~C–H!51.0606 Å and R~C–C!51.2032 Å;43 —for HCN, R~H–C!51.064 Å and R~C–N!51.156 Å,44 —for H2O, R~O–H!50.972 Å and ,HOH5104.5.12 The linear molecules are placed along the z axis, N2 and C2H2 symmetrically and HCN with the positive z direction from H to N. The water molecule has z axis as C2 axis and lies in the xz plane, with positive z direction from O to the H atoms. We discuss the shielding derivatives for the atoms located at (0,0,2z) in N2 and C2H2 and for the (1x,0,1z) H atom in H2O. D. MCSCF configuration spaces The active spaces are labeled here by the number of active orbitals in the different irreducible representations of the molecule, using only D 2h and its subgroups. Thus the notation (n 1 n 2 ...n 8 ) in D 2h indicates the number of active orbitals in symmetries ( s g p ux p uy d g s u p gx p gy d u ), respectively. In C ` v , the symmetries are ordered as ( sp x p y d ) and in C 2 v as (a 1 b 2 b 1 a 2 ). The core orbitals are inactive in all calculations. Since we are using a finite field technique, we could not exploit the full point group symmetry. Thus the CAS labels define the wave function for the unperturbed molecule. Symmetry reduction also means a large increase in the number of determinants: The largest CI expansion used includes over 800 000 determinants. The active spaces were selected based on the MP2 natural orbital occupation numbers. The first function used for N2 is a full valence CASSCF, ~21102110!. The second wave function includes five more orbitals in the active space, giving ~42203110!. For C2H2 the same CAS choices are made. For HCN we use only one CASSCF function, ~5220!. Because of lower symmetry, a wave function corresponding to the larger CAS of N2 would include more than 4.5 million determinants and the calculation was not attempted. For H2O the first function is a ~4220! CAS, the second a ~6331! CAS. When needed, we shall use the labels CAS A and CAS B to denote the wave functions with the smaller and larger active spaces, respectively. It has been observed that by using small complete active spaces one may overestimate the correlation corrections for the shielding.13 We have noticed a similar effect for the shielding polarizabilities and this is the reason we attempted to use, whenever possible, at least two different active spaces. IV. MAGNETIZABILITY POLARIZABILITIES The results obtained for the magnetizabilities and hypermagnetizabilities of the four molecules N2 , C2H2 , HCN, and H2O are summarized in Tables I to IV. Table V, where we have gathered results for all four molecules, displays the J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8959 TABLE I. N2 magnetizabilities x ab , hypermagnetizabilities h ab,cd and anisotropy Dh ~a.u.!. Here and in the following tables the sequence for the indices ab,cd in the hypermagnetizability tensor h is: a,b: electric field; c,d: magnetic field. See the text for details. SCF x xx x zz h xx,xx h xx,y y h xx,zz h zz,xx h zz,zz h xz,xz Dha Correlated Basis IVa Basis IVb Ref. 55 CAS B Basis IVa CAS B Basis IVb Ref. 55 ~MP2! 21.94 23.86 21.94 23.86 21.95 23.86 22.11 23.82 22.11 23.82 22.12 23.81 212.09 263.58 234.53 22.66 221.98 13.33 25.76 263.28 234.14 21.94 221.48 13.04 29.91 267.52 234.07 22.11 221.00 13.10 233.10 275.04 239.78 221.68 227.08 13.46 227.62 276.46 239.90 221.26 226.82 13.46 231.91 274.32 236.47 217.80 223.82 13.14 28.24 30.78 30.38 24.92 27.94 24.46 Experiment: Dh596.9674.6 a.u. ~Ref. 56!. a temperature dependence of the Cotton–Mouton constant, comparing with experiment and other reference calculated values. The calculation of molecular magnetizabilities has directly or indirectly been the subject of several studies. Recent SCF and correlated ~MP2! results for xab in N2 , HCN, and H2O with inclusion of vibrational corrections have been presented by Cybulski and Bishop.45 Their approach is not gauge origin independent, but the use of extended basis sets leads to a good overall reliability of the final results. The MC-IGLO method, an extension of the SCF-IGLO method used for studies on H2O,46 HCN, and C2H2 ,47 has been applied by van Wüllen and Kutzelnigg48 to N2 and H2O. The approach is multiconfigurational and gauge origin independent, in principle less computationally intensive than ours although somewhat dependent on the choice of the localization scheme for the molecular orbitals and on the completeness of the basis set. Sauer et al.44 have calculated magnetizabilities of N2 and HCN both at RPA ~resorting to Geertsen’s gauge independent approach49! and correlated ~SOPPA! levels of approximation. We also mention here the work of Jaszunski et al.42 with RPA and MCRPA results for N2 and Geertsen’s50 data for H2O. LAOs were employed within our group to compute magnetizabilities at the SCF level for a series of diamagnetic molecules including H2O.12 MCSCF magnetizabilities were also computed in an independent work.27 Notice finally that the only literature data we are aware of for C2H2 was obtained in our group using a gauge origin dependent approach.41 Very few studies of magnetizability polarizabilities have been published. Calculations are easier for atoms,33,34,51 due to the higher symmetry. H2 and D2 were studied by Fowler and Buckingham,52 benzene by Augspurger and Dykstra.53 Most of the work in this area has been carried out by Bishop and co-workers, who published correlated results for H2 ~D2!54 and very recently accurate SCF and MP2 results for the hypermagnetizabilities of H2 , N2 , HF, and CO.55 To our knowledge, there are no literature data for C2H2 , HCN and H2O. From the experimental point of view, the quantity most directly related to the hypermagnetizability is the Cotton– Mouton constant m C, which depends on the hypermagnetizability anisotropy according to Eqs. ~39! and ~40!. N2 and C2H2 are the only two molecules studied here for which experimental investigations aimed at measuring the hypermagnetizability anisotropy have been conducted.56,57 For these molecules the temperature dependent part, unrelated to the hypermagnetizability anisotropy, gives by far the most important contribution to m C. This is confirmed by our results and also by the calculations of Cybulski and Bishop55 for N2 . The experiment itself is extremely difficult, which adds to the uncertainty in the estimates of Dh. A. Magnetizability polarizabilities of N2 Table I shows the results for the N2 molecule. The two sets IVa and IVb give nearly the same results for both the parallel and the perpendicular components of the magnetizability tensor. Our numbers practically coincide with those of Ref. 45, both at SCF and correlated levels of approximation. Apparently, the basis set is saturated at the IVb level as far as the magnetizability is concerned. Our best average value for x ~22.68 a.u.! slightly improves our previous MCRPA estimate42 ~22.66 a.u.!. The MC-IGLO results for the tensor components are 23.76 a.u. ( x zz ) and 22.10 a.u. ( x xx ). All these studies show that correlation increases ~by about 5% in our case! the average magnetizability ~in absolute value!, essentially increasing the perpendicular component. This behavior is not reproduced by SOPPA.44 Convergence with respect to the basis set is not yet satisfactory for the magnetizability polarizabilities, at least for some of the components. In particular the perpendicular ( h xx,xx ) component is significantly modified going from IVa to IVb, both at the SCF and MCSCF levels. The SCF results are in good agreement with those of Ref. 45, with the exception of h xx,xx , where the result of Cybulski and Bishop ~29.91 a.u.! falls between our IVa and IVb results. The authors in Ref. 45 remark that the sum rules45,55 they use to verify the degree of gauge origin independence are ‘‘less well satisfied’’ for h ~especially for h xx,xx ! in N2 than in the J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8960 TABLE II. C2H2 magnetizabilities x ab , hypermagnetizabilities h ab,cd and anisotropy Dh ~a.u.!. See the text for details. SCF Basis IVb x xx x zz h xx,xx h xx,y y h xx,zz h zz,xx h zz,zz h xz,xz Dh c Correlated Basis IVa-CAS B 24.71a 25.21a 24.65b 25.06b 277.42 2205.34 2193.08 284.24 285.58 38.87 275.16 2202.72 2182.24 289.68 286.36 37.01 88.98 86.85 Reference values: 24.748 ( x xx ), 25.223 ( x zz ), Ref. 47 ~IGLO!. Reference values: 25.339 ( x xx ), 25.074 ( x zz ), Ref. 41 ~MCSCF, gauge origin in the center of mass!. c Experiment: 20654 ~Ref. 57. Notice the use of König’s definition for Dh, see the text! and 455634 ~Ref. 31!. a b TABLE III. HCN magnetizabilities x ab , hypermagnetizabilities j ab,c and h ab,cd and anisotropy Dh ~a.u.!. The sequence of the indices ab – c in j ab,c both here and in the next table is a,b: magnetic field; c: electric field. See the text for details. SCF Basis IVb Correlated Basis IVa x xx x zz 23.10a 24.45a 23.20b 24.35b j xx,z j zz,z j xz,x 5.08 0.06 0.90 4.88 0.38 1.08 229.90 298.82 279.40 22.38 242.38 22.86 233.44 2100.37 268.75 219.00 241.67 21.75 42.53 41.40 h xx,xx h xx,y y h xx,zz h zz,xx h zz,zz h xz,xz Dh a Reference values: 23.118 ( x xx ), 24.447 ( x zz ), Ref. 45, CHF. Reference values: 23.118 ( x xx ), 24.370 ( x zz ), Ref. 45, MP2. b other molecules, confirming that it is quite difficult to achieve basis set convergence for the h xx,xx component. The correlation effect is large, in particular for the electric field derivatives of x xx ~h xx,xx and h zz,xx !. As already pointed out in Ref. 45, the hypermagnetizability anisotropy Dh is less sensitive to the effect of correlation than these individual components. It decreases by about 10% in our CAS B IVb calculation ~27.94 a.u.! compared to the corresponding SCF value ~30.78 a.u.!. The best correlated value of Ref. 45 ~24.46 a.u.! is still lower by about 12%. The experiment gives a value of 96.9674.6 a.u.,56 leaving both us and Ref. 45 within the error bars! B. Magnetizability polarizabilities of C2H2 The magnetizability and magnetizability polarizabilities of C2H2 are reported in Table II. For this molecule, as for HCN and H2O, there are no theoretical results for h available for comparison. Moreover, as mentioned above, the only correlated numbers for x in the literature were obtained in our group as a byproduct of a calculation of the Verdet constant, with a [12s7p4d u 6s3p]/ ^ 6s5p4d u 4s3p & basis set, the CAS B wave function and using a gauge dependent approach.41 Our SCF magnetizability tensor elements and those obtained in the SCF-IGLO approach by Schindler and Kutzelnigg47 are in good agreement, as well as those by Ruud et al.58 We mentioned above that the differences between the components of h at the SCF level using the sets IVa and IVb are smaller than 5%, and that we cannot use IVb in the correlated calculations. Correlation effects are smaller in C2H2 than in N2 ~less than 4% on the average for the hypermagnetizabilities!. The anisotropy decreases by about 2%–3% ~86.85 a.u.! when correlation is introduced compared to SCF ~88.98 a.u.!. There are two very different experimental estimates for Dh of C2H2 : a recent 20654 a.u. value by Coonan and Ritchie57 and an older 455634 a.u. value by Kling et al.31 Our result disagrees with both experiments. It falls in between them, but it is much closer to the first. C. Magnetizability polarizabilities of HCN In HCN the first derivatives jab,g are nonzero, see Table III. For these derivatives, as well as for the second derivatives and Dh, we have no theoretical or experimental numbers to compare with. Our SCF magnetizability is in good agreement with that computed by Cybulski and Bishop.45 The average value ~23.55 a.u.! practically coincides with that of Ref. 45 and is close to 23.54 a.u. obtained by Sauer et al.44 The correlated average value ~23.58 a.u.! may be compared with 23.52 a.u. MP2 in Ref. 45 and the SOPPA value of 23.38 a.u. in Ref. 44. Apparently, Cybulski and Bishop45 find a slightly smaller effect of correlation ~none for x xx ! than we do, and in the opposite direction for the average value. All in all correlation plays a minor role for the magnetizability of HCN. Notice that we could not perform correlated calculations with basis set IVb and CAS B. Correlation plays a greater role for the magnetizability polarizabilities. In one case ( h zz,xx ) the effect is quite dramatic but, as for the two previous molecules, the influence on the anisotropy Dh is quite small ~less than 3%!. In axial molecules one obtains from Eq. ~41! Dh5 1 15 ~ 7 h xx,xx 25 h xx,y y 12 h zz,zz 22 h xx,zz 22 h zz,xx 112 h xz,xz ! ~43! which shows that the contribution of h zz,xx ~219.00 a.u.! to the anisotropy is, for instance, about an order of magnitude smaller than that of h xx,y y ~2100.37 a.u.!, which is only slightly influenced by correlation. J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties TABLE IV. H2O magnetizabilities x ab , hypermagnetizabilities j ab,c and h ab,cd and anisotropy Dh ~a.u.!. See the text for details. SCF Correlated Basis IVb Basis IVb-CAS A Basis IVb-CAS B x xx xyy x zz 22.91 22.97a 22.96a 22.99 23.02 23.03 22.97b 23.01b 23.02b j xx,z j y y,z j zz,z j xz,x j yz,y 20.33 0.45 0.08 20.22 20.22 20.20 0.69 0.31 20.43 20.36 20.22 0.67 0.31 20.40 20.33 h xx,xx h xx,y y h xx,zz h y y,xx h y y,y y h y y,zz h zz,xx h zz,y y h zz,zz h xy,xy h xz,xz h zy,zy 211.28 212.70 221.60 251.42 223.66 251.70 229.50 241.32 214.61 8.02 4.25 13.82 218.87 214.87 226.75 274.72 234.08 268.48 244.25 250.50 218.12 12.35 7.50 18.40 218.12 218.75 229.63 271.60 234.48 267.68 240.87 252.06 220.13 11.50 7.38 38.75 17.71 24.46 32.06 a Dh 8961 ing from set IVa to set IVb. It appears that we need a set larger than IVb to be able to claim basis set convergence. Therefore, we list in the table only IVb results. Both our SCF and CASSCF magnetizabilities are in good agreement with the results of Ref. 45. The average SCF magnetizability ~22.95 a.u.! compares well with other results: 22.93 a.u. ~Ref. 45!, 22.95 ~Ref. 48, SCF-IGLO!, and 22.88 ~Ref. 50, RPA!. Correlation gives xav.523.00 a.u., a value very close to the MP2 23.02 a.u. result of Ref. 45, the 22.925 a.u. value of Ref. 48 ~MC-IGLO! and the 22.963 a.u. CCPPA result of Geertsen.50 Correlation effects are quite important for the magnetizability polarizabilities. The correlated anisotropy ~32.06 a.u.! is almost twice the SCF value ~17.71 a.u.!. The effects on the individual tensor components, although never as dramatic as in some previous cases, are quite strong. As far as convergence with respect to the configuration space is concerned, the anisotropy increases by 25% going from CAS A to CAS B, a change closely related to the change in the h zy,zy component, which goes from 18.40 to 38.75 a.u. Apparently we are still not converged with respect to the correlation treatment. E. Temperature dependence of the Cotton–Mouton constant Reference values: 22.897 ( x xx ), 22.945 ( x y y ), 22.949 ( x zz ), Ref. 45, CHF. b Reference values: 22.992 ( x xx ), 23.041 ( x y y ), 23.033 ( x zz ), Ref. 45, MP2. a It is interesting to discuss the temperature dependence of the Cotton–Mouton constant m C and to compare with the literature, see Table V. Our references are here Eqs. ~39! to ~41!. Accordingly, we report in Table V our ‘‘best’’ results for Dh, Da, and Dx, comparing them with both experiment and other theoretical estimates. The quantity Q(T) is defined as D. Magnetizability polarizabilities of H2O Q~ T !5 Table IV summarizes the results for H2O. We have already mentioned that for this molecule changes of about 10% in the SCF hypermagnetizabilities were observed when go- 1 @~ a xx 2 a y y !~ x xx 2 x y y ! 1 ~ a y y 2 a zz ! 15kT 3 ~ x y y 2 x zz ! 1 ~ a xx 2 a zz !~ x xx 2 x zz !# , TABLE V. Temperature dependence of m C. Atomic units used unless explicitly specified. Q(T)51(2)/(15kT)( a i 2 a' )•( x i 2 x' ) for N2 , C2H2 , and HCN. i, j D a Q(T) 5 (1)/(15kT) @ ( a xx 2 a y y )( x xx 2 x y y ) 1 ( a y y 2 a zz )( x y y 2 x zz ) 1 ( a xx 2 a zz )( x xx 2 x zz ) # 5 (1)/(15kT) ( x,y,z ii, j j • D x ii, j j for H2O. N2 Our estimate Basis IVb CAS B Dh 27.94 Daa 4.37 Dxb 21.71 Q(273.15 K! m C(273.15 3 22 ~cm G K! mol21!31018 C2H2 Our estimate Basis IVa Ref. CAS B Others 24.46 96.9674.6 4.92 4.75160.088 55 56 61 56 86.85 21.69 21.85060.076 55 56 20.41 21151.83 24.22 24.4 23.92c 24.3c 23.86d 24.160.2d 11.29 HCN Others Ref. 20654 455634 12.5760.13 11.58 57 31 62 63 41.40 20.5060.03 20.7960.02 57 31 21.15 2713.49 59 56 60 22.36 22.17c Our estimate Basis IVa CAS A 8.14 21442.89 23.6160.2c 23.5260.15c 31 57 25.27 H2O Our estimate Basis IVb Others Ref. CAS B Others Ref. 32.06 9.04 21.25 64 45 0.44 0.79 0.36 0.049 0.042 20.007 3.98 0.41 0.68 0.27 0.049 0.041 20.008 65 45 0.14 ai2a' for the axial molecules. D a xx,y y , D a xx,zz and D a y y,zz for H2O. xi2x' for the axial molecules. D x xx,y y , D x xx,zz and D x y y,zz for H2O. c At 293.15 K. d At 298.15 K. a b J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8962 TABLE VI. N2 . Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. Here and in the following 8 and s ab,cd 9 is a: nuclear moment; b: magnetic field; c,d: tables the sequence of the indices ab-c(d) in s ab,c electric field. See the text for details. SCF 5 1 15kT ( i, j Correlated Basis IVa Basis IVb Ref. 73 Basis IVa CAS B Basis IVb CAS B Refs. 72 & 73 ~MP2! sav. Az B xx B zz 2109.4 21047.0 2297.0 1375.2 2109.4 21047.3 2297.8 1379.0 2111.38 21051.7 2294.7 1378.1 251.8 2837.6 2897.6 716.2 251.8 2837.9 2890.0 711.6 239.70 2777.0 21029.6 796.8 s xx s zz 8 s xx,z 8 s zz,z 8 s xz,x 8 s zx,x 9 s xx,xx 9 y s xx,y 9 s xx,zz 9 s zz,xx 9 s zz,zz 9 s xz,xz 9 s zx,xz 2333.5 338.6 1547.1 46.9 85.4 21402.8 21947.6 4564.2 24070.3 2834.4 2110.5 21130.0 2859.4 2333.4 338.6 1547.5 46.9 85.4 21403.0 21984.5 4602.3 24081.5 2831.0 2111.0 21127.5 2845.2 2336.4 338.6 1554. 46.92 85.81 21409. 22021. 4585. 24086. 2796.3 2111.0 21136. 2834.2 2247.4 339.5 1236.6 39.4 69.4 2653.6 1555.8 4813.0 22089.5 21002.1 2118.4 2898.4 2492.4 2247.4 339.5 1237.1 39.4 69.4 2652.4 1474.1 4865.0 22075.3 2999.2 2118.9 2896.4 2481.5 2229.5 339.8 1147. 36.86 72.35 235.28 2785. 4248. 22332. 2856.2 2119.4 2815.9 2269.8 D a ii, j j •D x ii, j j , ~44! x,y,z and it includes all the temperature dependence of m C, according to Eq. ~39!, for the molecules studied in this work. A comparison of Q(T) with the temperature independent part, Dh, shows that for the three axial molecules the hypermagnetizability anisotropy gives only a minor contribution to m C, compared to the contribution from the T-dependent factor: about 2.5% for N2 , 14% for C2H2 and 3% for HCN at 273.15 K. In all three cases Dh is of opposite sign with respect to Q(T). For C2H2 , the most recent experiment predicts 22%66% relative contribution of the T-independent term to m C. 57 The data furnished by Kling and Hüttner56 and reported in Table V can be combined to predict approximately a 7%65.5% contribution for N2 . H2O shows a completely different behavior. Our results give Q(T)53.98 a.u. at 273.15 K, while our best value for the hypermagnetizability anisotropy is 32.06 a.u. In this case the temperature dependent term contributes a little more than 4% to m C. The water molecule thus shows a ‘‘quasiatomic’’ response to the electromagnetic perturbation as far as the Cotton–Mouton effect is concerned: the predicted value of 219 cm3 G22 mol21 at 273.15 K, is quite close to m C, 1.4310 that computed for argon33 ~9.1310220 cm3 G22 mol21, independent of T! and about an order of magnitude larger than that obtained for isoelectronic neon atom34 ~1.0310220 cm3 G22 mol21, again independent of T!. In Table V we also compare the calculated values of C with experiment for N2 and C2H2 . N2 is a molecule of m special importance in experiments, since it has been used as a reference system for determinations of Cotton–Mouton constants in atomic and molecular systems59 and for calibration of the experimental apparatus.60 Our values of m C of N2 at different temperatures agree quite well with experiment. There is some disagreement, however, between our values and experiment for m C in C2H2 . We have already discussed our limitations and the factors we believe are the sources of uncertainty in our calculation. For atoms the calculated values of m C are accurate to a few percent.33,34,51 Although the accuracy of ab-initio calculations in molecules like C2H2 is lower, we do not believe that the errors in the computed electric polarizability and magnetizability anisotropies can explain the discrepancy ~up to 40%! with experiment, and in view of the relatively minor contribution given by the temperature independent factor to m C, the overall disagreement can hardly arise from our uncertainty in Dh. V. SHIELDING POLARIZABILITIES There have been numerous calculations of the shielding tensors for all the nuclei in N2 , C2H2 , HCN, and H2O. For a recent review of various methods used in shielding tensor calculations and references to earlier results, see Ref. 66. The same methods which were used for magnetizabilities have also been applied to compute the shielding constants in these four molecules. There are IGLO47 and MC-IGLO,48 RPA, and SOPPA,44,67 SCF-GIAO,68 and MCSCF-GIAO13,69 results for most of the nuclei in these molecules. In addition, there are MCSCF42 and MP270 values, and recent MBPT15 and coupled cluster ~CCSD!71 calculations using GIAO basis sets. We shall not discuss these results in such detail as the corresponding magnetizability values. There are few other results for magnetizability derivatives. In contrast, there are literature data for all the shielding polarizability tensors we compute and we can compare our results directly to these J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8963 TABLE VII. C2H2 . Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See the text for details. C atom H atom SCF 115.7 B xx B zz 146.6 556.6 s xx s zz 8 s xx,z 8 s zz,z 8 s xz,x 8 s zx,x 9 s xx,xx 9 y s xx,y 9 s xx,zz 9 s zz,xx 9 s zz,zz 9 s xz,xz 9 s zx,xz a 119.772 119.8 2750.7 2733.9 116.4 583.1 553.45 40.246 278.825 1097.8 56.5 2756.2 Az 34.3 278.5 1106.0 56.5 242.0 2212.1 2652.1 2689.7 21546.5 23144.0 2246.8 473.7 21129.0 SCF Basis IVa CAS B Othersa Basis IVb sav. Correlated 2542.0 2659.0 21619.8 22815.5 2258.8 30.4 2639.6 269.1 146.9 675.6 146.6 9.0 54.1 279.1 934.9 49.1 232.7 262.3 2677.6 1517.9 21906.7 22694.7 2240.0 455.6 21336.0 25.1 40.9 73.7 60.0 226.8 232.7 2135.0 2507.6 20.6 2236.9 252.8 2163.5 2177.3 Basis IVa CAS B Othersb Basis IVb 129.1 Correlated 30.76 33.7 270.1 267.2 71.3 2191.0 6.8 25.73 40.82 76.3 57.8 30.5 265.6 146.9 25.4 25.3 40.9 70.7 55.4 226.3 220.2 2154.4 2491.7 238.6 2235.5 275.0 2159.1 2147.5 226.5 2428.5 552.0 2226.0 101.8 First number: Ref. 77. Second ~if present!: Ref. 75. First number: Ref. 39. Second ~if present!: Ref. 75. b reference values. Therefore, we discuss primarily the shielding constants obtained in the studies that included at the same time shielding polarizabilities, and these results are shown in the tables. We refer to Refs. 16 and 75 for more data on accurate ab initio and experimental results for s. For all the nuclei in all four molecules our and other recent SCF calculations yield similar results. The differences at the SCF level are usually related to differences in assumed geometries. The correlation corrections for s~N! in N2 are large and fairly well described by our MCSCF wave function. For HCN, the CAS A function overestimates the correlation effects, while for C2H2 and H2O they are not so large and the comparison does not really help us estimate the accuracy of TABLE VIII. HCN. Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See the text for details. Atom H SCF Basis IVb sav. Az B xx B zz s xx s zz 8 s xx,z 8 s zz,z 9 s xx,xx 9 y s xx,y 9 s xx,zz 9 s zz,xx 9 s zz,zz a 29.2 255.9 69.1 80.4 24.3 39.0 57.9 51.9 246.7 2267.8 2227.2 299.8 227.9 Atom C Correlated a Others 31.97 32.7 254.9 254.1 26.9 66.2 86.6 25.09 39.12 56.4 51.8 157.0 2217.0 2184.3 2101.5 228.8 Basis IVa SCF Basis IVb 29.2 71.1 255.2 2440.2 48.5 76.3 2283.3 2743.5 24.2 39.2 57.6 50.6 2.5 2190.9 2221.5 2102.6 215.1 232.1 277.5 637.8 45.1 21039.0 3885.9 2293.9 21147.2 2126.8 Atom N Correlated b Others 75.746 76.3 2428.6 2422.6 2290.8 2744.8 2751.45 225.287 277.812 620.3 45.3 2998.5 3855.5 2299.3 21112.5 2129.8 SCF Basis IVa Basis IVb 93.8 250.4 2418.7 1949.1 142.1 2513.5 687.6 5680.6 1.5 278.3 608.6 38.9 21137.3 1756.7 1616.1 21472.2 2151.0 2245.2 339.3 22895.1 257.2 2420.4 21037.7 216943.8 22667.6 2195.9 Correlated c Others 245.03 245.0 1927.0 1910.1 651.2 5667.0 5668.05 2237.29 339.48 22861.9 257.3 2408.0 21162.0 216901.2 22337.0 2199.5 Basis IVa 17.5 1434.9 2581.4 3551.0 2143.7 340.1 22128.9 246.8 1576.5 3629.0 210587.1 21716.8 2132.1 First number: Ref. 39. Second ~if present!: Ref. 75. First number: Ref. 77; Second ~if present!: Ref. 75. First number: Ref. 40; Second ~if present!: Ref. 75. b c J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties 8964 TABLE IX. H2O. Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See text for details. O atoma H atom SCF sav. Ax Az B xx Byy B zz B xz s xx syy s zz 8 s xx,x 8 s xx,z s 8y y,x s 8y y,z 8 s zz,x 8 s zz,z 9 s xx,xx 9 y s xx,y 9 s xx,zz s 9y y,xx s 9y y,y y s 9y y,zz 9 s zz,xx 9 y s zz,y 9 s zz,zz 9 s xx,xz s 9y y,xz 9 s zz,xz Correlated b c Our geometry Geo. Ref. 40 Others 320.5 0.0 2437.1 3582.3 2653.7 2017.5 0.0 358.9 297.7 304.9 0.0 605.0 0.0 320.8 0.0 385.3 25212.6 2716.0 211375.5 28285.0 834.6 2145.1 27996.3 3803.5 2584.6 0.0 0.0 0.0 328.4 0.0 2378.2 3002.8 2547.8 1767.0 0.0 366.4 305.5 313.3 0.0 513.0 0.0 280.9 0.0 340.7 24282.0 2908.8 210068.2 27122.5 791.4 2142.3 26612.3 3404.1 2391.8 0.0 0.0 0.0 329.06,325.3 0.0 2381.6,2401.1 2946.1 2363.3 2074.0,2169.65 0.0 366.78 306.15 314.24 0.0 518.5 0.0 284.3 0.0 342.1 24182.0 21232.5 211045.0 27000.5 458.5 2388.1 26494.0 2954.0 21010.8 0.0 0.0 0.0 SCF CAS A CAS B Our geometry Geo. Ref. 40 338.3 0.0 2306.4 1845.9 2806.7 1253.5 0.0 372.1 315.0 327.9 0.0 455.5 0.0 220.0 0.0 243.8 23558.9 21064.0 29288.7 25247.6 1099.3 715.6 22268.9 4805.1 1051.8 0.0 0.0 0.0 335.3 0.0 2336.7 2125.2 2800.0 1367.6 0.0 367.7 312.6 325.6 0.0 521.2 0.0 235.0 0.0 254.0 23748.7 2950.5 29866.3 25723.3 1104.0 768.6 23278.9 4646.5 891.8 0.0 0.0 0.0 30.1 79.6 43.5 262.6 96.0 24.8 2273.7 37.9 22.8 29.7 256.1 250.3 287.4 253.6 295.1 226.6 23.2 2190.7 2.7 229.0 2131.6 271.1 149.6 2253.5 280.2 301.9 250.1 269.2 30.8 76.6 42.7 255.9 96.4 24.7 2254.5 38.7 23.2 30.3 253.2 249.7 285.1 252.5 291.5 225.8 24.8 2195.1 2.9 207.2 2129.4 270.1 132.8 2254.1 281.1 280.6 238.8 244.1 Correlated b Others d 32.34,36.7 78.5 47.2,47.3 243.9 97.2 6.1,40.2 2262.4 39.52 25.22 34.59 CAS A CAS B 30.2 73.3 42.0 229.3 128.3 42.5 2212.7 38.0 22.9 29.7 251.3 250.6 283.3 250.8 285.1 224.7 223.3 2277.8 212.5 137.9 2151.7 2117.0 61.5 2340.1 2125.7 258.4 213.6 166.1 30.2 74.9 42.2 231.0 126.9 45.5 2223.9 38.0 22.9 29.7 252.5 250.5 284.6 251.4 287.4 224.8 229.1 2271.2 223.2 144.5 2154.0 2124.3 70.8 2336.5 2125.6 271.5 220.8 179.5 a 0.0 values exact by symmetry. Bond lengths and angle as in Ref. 40. c First number: Ref. 40. Second ~if present!: Ref. 75. d First number: Ref. 74 ~recomputed rotating the axes!. Second ~if present! Ref. 75. b the computed shielding derivatives. The strong geometry dependence of the shielding constants makes this comparison difficult, in particular when various data do not differ significantly. For the shielding polarizabilities, there are few experimental results, none directly related to the values calculated here. We refer to the works of Grayson and Raynes39,40,74 for a discussion of experimental data. In the tables we use the following definitions for the ‘‘mean’’ shielding polarizability tensors ~Einstein summation assumed!: A z 52 1 3 8 ,z , s aa B zz 52 1 6 9 ,zz , s aa ~45! and similarly for the other components. These definitions are the same as those used by Bishop and co-workers ~see, for instance, Refs. 72 and 73! and by Raynes and collaborators ~see, e.g., Refs. 39 or 74!, but differ for B from Ref. 75 ~the quoted values are hence scaled by a factor of 20.5 compared to the values of Ref. 75!. A. Shielding polarizabilities of N2 Table VI summarizes the results for the nuclear magnetic shielding and shielding polarizabilities for N2 . The SCF results for all components are in fairly good agreement with the data of Bishop and Cybulski.73 The largest difference ~for 9 ! is less than 5%, for the other components the differs zz,xx ences are of the order of 1%–2% or less. The correlation corrections computed with the CAS B wave function are in general in good agreement with the MP2 values of Refs. 72 and 73. Only the final results for the x field derivatives differ significantly from those of Bishop and Cybulski.72,73 However, the correlation correction has the same sign for all derivatives. For these components, the CAS correlation corrections are about 60% of the MP2 values, e.g., for s 9xx,xx both SCF values are about 22000, our final result is 11494 and Bishop and Cybulski obtain 12785. It is not clear at this stage whether CAS underestimates or MP2 overestimates the correlation effect. The value of s~N! itself computed in the MCSCF approximation using large active space, agrees better than MP2 with experiment, which gives 261.660.2 ppm for s~N! in N2 .16,70,76 B. Shielding polarizabilities of C2H2 Table VII presents the results for C2H2 . Our SCF values for s~C! and its derivatives agree well with the literature ~see also Refs. 47 and 68!. The correlation corrections to s8 and s9, although significant, are much smaller than for N2 . Also J. Chem. Phys., Vol. 102, No. 22, 8 June 1995 Downloaded 07 Jun 2013 to 193.157.137.211. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Rizzo et al.: Electric field dependence of magnetic properties for the A and B coefficients the SCF approximation yields a reasonable value. This reflects the fact that for C2H2 , the SCF values for s~C! and s~H! are, in contrast to s~N! in N2 , close to the correlated ~and experimental! results. C. Shielding polarizabilities of HCN The HCN results, summarized in Table VIII, are similar to those for N2 and C2H2 . The correlation corrections are most important for the N atom, starting with the change in s~N!. As mentioned above, due to lower symmetry of HCN, we were not able to use a CAS B active space. The accuracy of the correlated results for s thus is lower. CAS A overestimates the correlation effects for s~C! and s~N!.71 Presumably, the same applies to the correlation corrections to the shielding derivatives. Our SCF results differ significantly from literature SCF results39,75 for some components of s9~H! in C2H2 and HCN. Calculations placing the gauge origin at the H atom are very demanding and this is probably the reason we observe for the H atom larger differences than for other atoms. As we use LAOs and have employed larger basis sets than those of Refs. 39 and 75, we believe that our SCF results are more reliable. D. Shielding polarizabilities of H2O In Table IX, reporting the results for H2O, we have included only the nonzero derivatives of the diagonal tensor components of s. We do not discuss the off-diagonal components and their derivatives. Basis IVa and IVb give the same results for SCF. In Table IX we have shown the results for CAS A and CAS B wave functions. As for other molecules, CAS A overestimates the correlation corrections.71 For the oxygen shielding derivatives, where the correlation corrections are most significant, the CAS A results provide a fair approximation to CAS B, but the differences are noticeable. We have not analyzed systematically the dependence of the properties on the geometry. For N2 , we refer to Bishop and Cybulski72 for a careful study of the geometry dependence and rovibrational effects. We have noticed, however, that our SCF results for shielding polarizabilities in H2O differ significantly from those of Grayson and Raynes.40 Therefore, we have repeated these SCF calculations using the geometry of Ref. 40 and as shown in Table IX the differences become much smaller. This indicates that the geometry dependence of the shielding polarizabilities is quite strong, which is not surprising considering the large changes in s~O! with molecular geometry. VI. CONCLUSIONS For many of the properties studied in this work, our calculations represent the first analysis going beyond the SCF approximation. The calculations confirm the strong dependence of these properties on the basis set and correlation corrections. It appears that for small molecules at the SCF level it is possible to saturate the basis set. It is much more difficult to reach convergence in the treatment of correlation effects. 8965 A mixed analytic-numerical differentiation technique was used to compute the hypermagnetizabilities and shielding polarizabilities. The calculations, although time consuming, are feasible. When the first derivatives ~third-order properties! are non zero, the calculations appear to be rather accurate. For second derivatives the accuracy is lower, and the allperpendicular tensor components for linear molecules are particularly difficult to describe. For these components, correlation effects may change both the sign and the order of magnitude of the SCF results. Considering the size of the correlation corrections to the second-order magnetic properties in the molecules we have studied, this is not very surprising. However, it shows that an accurate and reliable calculation of magnetizability and shielding derivatives requires the use of very large basis sets, a method which ensures gauge origin invariance and, at the same time, a wavefunction that describes all the important correlation effects. ACKNOWLEDGMENTS The authors wish to express their thanks to W. T. Raynes for kindly providing preprints of his recent work and for his comments on this manuscript. A major part of this work has been done during a visit of A. B., M. J., and A. R. to Aarhus. This work has been partly supported by the Danish National Research Council ~Grant No. 11-0924!. A. D. Buckingham, Adv. Chem. Phys. 12, 107 ~1967!. A. D. Buckingham and B. J. Orr, Quart. 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