Electric field dependence of magnetic properties: Multiconfigurational self-consistent field calculations of hypermagnetizabilities and nuclear shielding polarizabilities of N

Electric field dependence of magnetic properties: Multiconfigurational self-consistent field calculations of hypermagnetizabilities and nuclear shielding polarizabilities of N

, C

H

, HCN, and H

O

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”Jaszun´ski

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 par- ticular, 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 prop- erties 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 calcu- lation of magnetic properties, such as magnetizabilities and NMR shielding constants, are reflected in the unphysical de- pendence of the calculated values on the chosen gauge ori- gin. Even at the SCF level, it is practically impossible to ensure gauge origin independence of the results for a poly- atomic 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 meth- ods that enable practical use of these orbitals have only re- cently been formulated, first for SCF functions⁷ and more recently for arbitrary approximate wave function.⁸ The theory of linear response for multiconfigurational self- consistent field~MCSCF!wave functions⁹has been success- fully combined with the use of LAOs.^10–13 Nuclear shield- ings using London orbitals have also been calculated at second, third, and partly fourth^{14 –16} order in perturbation

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,^18,19ap- plying what is called the natural connection for perturbation- dependent basis sets. This is needed in order to avoid nu- merical 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 depen- dence 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 con- stant magnetic field. Isotropic substances show weak bire- fringence 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_i2n_'5C_CMlB²

linking the observed birefringence ~anisotropy of the refrac- tion 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

and uniform magnetic field. The resulting magneto-optic ef- fect 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 interac- tion with a distant part of the molecule ~see, e.g., Ref. 23!. We refer to some recent reviews^4,24,25 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 shieldings(K), defined as

x52]²e~B,m!

]^2B

U

s~K!511]²e~B,m! ]^B]^mK

U

K50

, ~2!

wheree~B,m!is the molecular energy in the presence of the external magnetic field induction B and the nuclear magnetic moments m. m_K is the nuclear magnetic moment of the Kth nucleus.

Switching on the electric field perturbation E, the ele- ments of the molecular magnetizability tensor x_ab ^{may be} written as

x_ab^E 5xab1jab,gE_g1¹2hgd,abE_gE_d1••• , ~3! where the hypermagnetizability tensors jab,g andhab,gd are given as

jab,g5]x_ab^E

]^E_g ^{, ~4!}

hab,gd5 ]²a_ab^B ]^Bg]^B_d⁵

]²x_gd^E

]^Ea]^Eb. ~5! In the first definition ofhab,gdthe electric dipole polarizabil- ity tensor a_ab 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 calcu- lated magnetizabilities.

Similarly, the shielding tensor for nucleus K may be ex- panded as

s_ab^E ~K!5s_ab~K!1s_ab8 ,g~K!E_g

1¹2 s_ab,gd9 ~K!E_gE_d1••• , ~6! where the so-called shielding polarizabilities

s_ab8 ,g~K!5]s_ab^E ~K!

]^Eg ~7!

and

s_ab9 ,gd~K!5]²s_ab^E ~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 observ- ables. 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!

H51 2

(

i

pi 22

(

i,K

Z_K r_i,K11

2

(

iÞj

1

r_{i j}, ~9! where the kinetic momentum of the ith electron is given as

pi52i“i1A~r_i! ~10! and the dependence on the magnetic fields is collected in the vector potential

A~r!51

2 B3r_O1a²

(

K

m_K3r_K

r_K³ , ~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 mag- netic fields this can be achieved by introducing the London atomic orbitals⁵

v_m~r_M;A_M!5e^~2iAMe–r!x_m~r_M!, ~12! where A_M^e represents the potential at the position of the nucleus M

A_M^e5¹2 B3R_{M O} ~13! andx_mis 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 one- electron, one-center system.¹¹

As shown by Helgaker and Jo”rgensen,⁸ the Hamiltonian integrals that occur for London atomic orbitals can be written in the following form:

^v_m~A_M^e!u1

2 p²2

(

K

Z_K

r_K uv_n~A_N^e!&

5^xmue^{iAM Ne –r}

S

⁽

D

~14!

g_mnrs~A!5^xmx_nuexp~iA_{M N}^e –r₁1iA_RS^e –r₂!

r₁₂ ux_rx_s&^{, ~15!}

where the new vector potentials are defined as

A_{M N}^e 5A_M^e2A_N^e5¹2B3B_{M N}, ~16! A~r!2A_N^e51

2 B3r_N1a²

(

K

m_K3r_K

r_K³ . ~17! 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 mag- netic 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 d_e and the external electric field as

H^int52E–d_e, ~18!

where d_e is defined as d_e52

(

i

r_i. ~19!

As the operator in Eq. ~18! commutes with the exponential operator in the London atomic orbital, the one-electron inte- grals we need can be written as

h_mn~A!5^xmue^{~iAM Ne –r!}

S

2

(

K

Z_K

r_K2E–d_e

D

where we have added the electric field interaction integral to Eq.~14!. No modification is needed in the expression for the two-electron integrals.

We now follow closely the exposition in a recent paper on MCSCF magnetizabilities using London atomic orbitals.²⁷Only the major steps are given, with emphasis on the differences arising when the electric field effects are in- vestigated.

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 pa- rameters in the reference wave function optimized at zero field and zero nuclear magnetic moments

uWF~B,m!&^5exp~ik!exp~iS!uRWF&^{. ~21!}

Our reference wave function is optimized at a given electric field E and constructed as a linear combination of Slater determinants

uRWF&⁵

(

b

C_bub&^{. ~22!}

The orbital and configuration operators kand S in Eq. ~21! describe magnetic perturbations and are therefore given by

k5

(

r.s

krs

R~E_rs1E_sr!, ~23!

S5

(

KÞ0

S_K^R~uK&^^RWFu1uRWF&^^Ku!. ~24!

Only nonredundant rotations are included in the summation of Eq.~23!. The one-electron excitation operators in Eq.~23! are defined as

E_mn5

(

s561/2

a_m¹_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 or- dered products of creation operators

ub&5

)

i

a_b

i

1uvac& ~26!

creating electrons in a space of orthonormalized molecular orbitals~OMOs!

fm

OMO~B!5

(

n

fn

UMO~B!T_nm~B!. ~27!

The unmodified molecular orbitals~UMOs!fn

UMO(B) corre- spond to the optimized orbitals at B50 and m50 and are linear combinations of London atomic orbitals

fn

UMO~B!5

(

m

C_nmvm~B!. ~28!

8955 Rizzo : Electric field dependence of magnetic properties

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

T5W²¹~WS²¹W¹!^1/2, ~29! where S is the UMOs overlap matrix

S_mn~B!5

(

mn

C_mmC_nn^vm~B!uvn~B!& ^~30!

and the nonsymmetric matrix W is given by W_mn~B!5

(

mn

C_nnC_mm^vn~B₀!uvm~B!&

5

(

mn

C_nnC_mm^xnuvm~B!&^. ~31!

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 in- volves 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 pro- vides 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,⁸can be written as

H~B!5

(

mn

h˜_mn~B!E_mn11

2

(

mn pq

g˜_{mn pq}~B!e_{mn pq}, ~32!

where the molecular integrals and two-electron excitation operator are defined as

h˜_mn~B!5

(

m8ⁿ8 T_mm

* 8_~B!hm

8ⁿ8~B!T_n8n~B!, ~33!

g˜_mnrs~B!5

(

m8ⁿ8^r8^s8 T_mm

* 8_~B!T_nn

*8_~B!g_m8n8r8s8~B!

3T_r8r~B!T_s8s~B!, ~34!

e_{mn pq}5E_mnE_pq2dpnE_mq. ~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

e~B,m!5^^RWFuexp~2ik!exp~2iS!H exp~iS!

3exp~ik!uRWF&^. ~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 ob- tained 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 inter- action operator. Thus the first and second derivatives of the one-electron Hamiltonian look like

]^hmn

]^B

U

51

2 ^xmuL_Nux_n&^1iQM N^xmurhux_n&

2iEQ_{M N}^xmurduxn&^, ~37!

]^2h_mn

]^B2

U

1Q_{M N}^xmurr^Thux_n&^QM N

2EQ_{M N}^xmurr^Tduxn&^QM 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 symme- trized, L_N is the operator for angular momentum around nucleus N, L_N52ir_N3“, 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 two- electron 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 in- dependent magnetizabilities and shielding constants has been presented. The shielding polarizabilities and hypermagnetiz- abilities can be obtained by calculations of the nuclear

shieldings and magnetizabilities at different electric field strengths, followed by differentiation with respect to the electric field strength.

C. Relationship to experiment

Before concluding this section, we give some definitions needed to relate the quantities we compute to experiment, in particular the hypermagnetizability hto the Cotton–Mouton effect. The first theoretical treatment of the Cotton–Mouton effect appeared in 1910.²⁸A fundamental contribution to the understanding of the topic was given by Buckingham and Pople in 1956.²⁹A classical statistical mechanics approach³⁰ shows that for rigid diamagnetic molecules the so-called Cotton–Mouton constant_mC, proportional to the mean elec- tric polarizability in a strong magnetic field, can be written as

mC52p^N

27

H

S

D

1 1

5kT

S

DJ

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

mC52p^N

27

F

G

whereDa5a_i2a_'^,Dx5x_i2x_', and the hypermagnetizabil- ity anisotropy is defined as

Dh5 ¹5 ~hab,ab2¹3 haa,bb! ~41! thus differing by a factor of five from that used by Hu¨ttner and collaborators.³¹Notice also that some authors~e.g., Ref.

31! employ Ko¨nig’s³² definition of_mC, a factor of 9 larger than Eq. ~39! above. Our results for _mC and Dh ^{are given} according to Eqs.~39!to~41!above.

The experimentally measured frequency dependent hy- permagnetizability can be written in terms of a combination of quadratic~diamagnetic contribution!and cubic~paramag- netic contribution!terms ~r position, L angular momentum, and Q quadrupole moment operators!³³

hab,gd~2v^;v^,0,0!5h_abp ,gd1h_abd ,gd

52¹2 ^^^ra;r_b,L_g,L_d&&2v;v,0,0

2 ¹4 ^^^ra;r_b,Q_gd&&2v;v,0. ~42!

By taking advantage of symmetry and thus reducing the ex- pression for the paramagnetic contribution to a Cauchy mo- ment expansion, we were able to study the frequency depen- dence of the hypermagnetizability anisotropy in the Ne and Ar atoms using the multiconfigurational response method.^33,34The same approach cannot straightforwardly be employed for molecules.

Atomic units are used throughout the paper~unless oth- erwise stated!. Conversion factors to some of the units used by other authors are given in the following:

—1 a.u. ofa5(e²a₀²)/(E_h)51.648 78310²⁴¹

C²m²J²¹ ~SI, MKSA!51.481 847310²²⁵ ~4pe0! cm³

~emu!;

—1 a.u. ofx5(e²a₀²)/(m_e)57.891 04310²²⁹J T²²or 7.891 04310²³⁰ cm³;

—1 a.u. ofj5(e³a₀³)/(m_eE_h)51.534 562310²⁴⁰ C m T²²51.379 196310²²⁶~4pe0!cm⁴s²¹G²¹;

—1 a.u. ofh5(e⁴a₀⁴)/(m_eE_h²)52.984 25310²⁵² C²m²J²¹T²²52.682 108310²⁴⁴~4pe0!cm³G²²;

—1 a.u. ofs85p pm(a₀²/e)51.944 69310²¹⁸ mV²¹55.830 03310²¹⁴ cm statV²¹~esu!;

—1 a.u. ofs95p pm(a₀²/e)²53.781 82310²³⁰ m²V²²53.398 92310²²¹ cm²statV²².

In Eq.~40!above the factor relating _mC to the quantity in square brackets ~computed in a.u.! is 3.758 738 10²²¹ cm³G²²mol²¹ ~4pe0! ~CGSM! or 5.935 561 10²³¹ m⁵A²²mol²¹~SI!.

The relationship between the experimental Cotton–

Mouton constant²¹C_CMand the_mC introduced by Bucking- ham and Pople,²⁹at a temperature T, 1 atm and in the CGSM unit system, is

C_CM~cm²¹G²²)5 27 2V_m• 1

l~cm! ^mC~cm³G²²mol²¹! 5 0.164 518

T~K!•l~cm! ^mC~cm³G²²mol²¹!, 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 magnetiz- ability derivatives~where the first derivative may vanish be- cause 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 Con- vex 3840 computer.

The convergence of the properties with respect to exten- sion 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 com- pute than others. We shall, therefore, describe in the text the

8957 Rizzo : 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 num- bers for the best SCF and best correlated calculation. To illustrate the basis set dependence we quote two sets of val- ues 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 includesHERMIT³⁵for the integrals,SIRIUS³⁶

for the wave function andABACUS³⁷ for the magnetic prop- erties.

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 thes9 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 N₂and C₂H₂we 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 ]/^^{8s7 p3d1 f}& contractions for C, N, and O atoms and a [6s3 p1d]/^^{5s3 p1d}& 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 electric- field 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,⁴¹we 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 ]/

^10s9 p5d2 f& functions for C, N, and O and [8s5 p2d]/

^^{7s5 p2d}& 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 proper- ties 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 SCFh ^compo- nents computed using basis sets IVa and IVb are smaller than 5% ~we exclude in these comparisons the very small tensor components!, fors8they do not exceed 1%, and fors9^10%.

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% forh^ands9at the SCF level, and significantly larger fors9at 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, N₂and 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 N₂and C₂H₂and 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 (n₁n₂...n₈) in D_2h indicates the number of active orbitals in symmetries (sgpuxpu ydgsupgxpg ydu), respec- tively. In C_`v, the symmetries are ordered as (spxpyd^{) and} in C_2v as (a₁b₂b₁a₂).

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 re- duction also means a large increase in the number of deter- minants: 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 N₂is a full valence CASSCF,

~21102110!. The second wave function includes five more orbitals in the active space, giving~42203110!. For C₂H₂the 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 N₂would 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 hyper- magnetizabilities 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

temperature dependence of the Cotton–Mouton constant, comparing with experiment and other reference calculated values.

The calculation of molecular magnetizabilities has di- rectly or indirectly been the subject of several studies. Recent SCF and correlated~MP2!results forxab in N₂, HCN, and H₂O with inclusion of vibrational corrections have been pre- sented 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 ap- plied by van Wu¨llen and Kutzelnigg⁴⁸ to N₂ and H₂O. The approach is multiconfigurational and gauge origin indepen- dent, in principle less computationally intensive than ours although somewhat dependent on the choice of the localiza- tion scheme for the molecular orbitals and on the complete- ness of the basis set. Sauer et al.⁴⁴ have calculated magne- tizabilities 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 inde- pendent work.²⁷Notice 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,^33,34,51due to the higher symmetry. H₂and 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.⁵⁵To 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_mC, which depends on the hypermagnetiz- ability anisotropy according to Eqs. ~39! and ~40!. N₂ and C₂H₂ are the only two molecules studied here for which ex- perimental investigations aimed at measuring the hypermag- netizability anisotropy have been conducted.^56,57 For these molecules the temperature dependent part, unrelated to the hypermagnetizability anisotropy, gives by far the most im- portant contribution to _mC. This is confirmed by our results and also by the calculations of Cybulski and Bishop⁵⁵for N₂. The experiment itself is extremely difficult, which adds to the uncertainty in the estimates ofDh^.

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 magnetiz- ability 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 im- proves our previous MCRPA estimate⁴² ~22.66 a.u.!. The MC-IGLO results for the tensor components are 23.76 a.u.

(xzz) and22.10 a.u. (xxx). All these studies show that cor- relation 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 sat- isfactory for the magnetizability polarizabilities, at least for some of the components. In particular the perpendicular (hxx,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 excep- tion of hxx,xx, where the result of Cybulski and Bishop

~29.91 a.u.!falls between our IVa and IVb results. The au- thors in Ref. 45 remark that the sum rules^45,55 they use to verify the degree of gauge origin independence are ‘‘less well satisfied’’ forh~especially forhxx,xx!in N₂than in the

TABLE I. N2magnetizabilitiesxab, hypermagnetizabilitieshab,cdand anisotropyDh~a.u.!. Here and in the following tables the sequence for the indices ab,cd in the hypermagnetizability tensorhis: a,b: electric field;

c,d: magnetic field. See the text for details.

SCF Correlated

Basis IVa Basis IVb Ref. 55

CAS B Basis IVa

CAS B

Basis IVb Ref. 55~MP2!

xxx 21.94 21.94 21.95 22.11 22.11 22.12

xzz 23.86 23.86 23.86 23.82 23.82 23.81

hxx,xx 212.09 25.76 29.91 233.10 227.62 231.91

hxx,y y 263.58 263.28 267.52 275.04 276.46 274.32

hxx,zz 234.53 234.14 234.07 239.78 239.90 236.47

hzz,xx 22.66 21.94 22.11 221.68 221.26 217.80

hzz,zz 221.98 221.48 221.00 227.08 226.82 223.82

hxz,xz 13.33 13.04 13.10 13.46 13.46 13.14

Dh^{a 28.24 30.78 30.38 24.92 27.94 24.46}

aExperiment:Dh596.9674.6 a.u.~Ref. 56!.

8959 Rizzo : Electric field dependence of magnetic properties

other molecules, confirming that it is quite difficult to achieve basis set convergence for thehxx,xxcomponent. The correlation effect is large, in particular for the electric field derivatives ofxxx~hxx,xxandhzz,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 forh^available for comparison. Moreover, as mentioned above, the only cor- related numbers forx in the literature were obtained in our group as a byproduct of a calculation of the Verdet constant, with a [12s7 p4du6s3 p]/^^{6s5 p4d}u4s3 p& 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 ofhat 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 C₂H₂than in N₂~less than 4% on the average for the hyper- magnetizabilities!. The anisotropy decreases by about 2%–3% ~86.85 a.u.! when correlation is introduced com- pared to SCF~88.98 a.u.!. There are two very different ex- perimental estimates for Dh ^{of C}2H₂: a recent 20654 a.u.

value by Coonan and Ritchie⁵⁷ and an older 455634 a.u.

value by Kling et al.³¹Our result disagrees with both experi- ments. It falls in between them, but it is much closer to the first.

C. Magnetizability polarizabilities of HCN

In HCN the first derivativesj_ab,gare nonzero, see Table III. For these derivatives, as well as for the second deriva- tives and Dh, we have no theoretical or experimental num- bers 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 to23.54 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 Bishop⁴⁵find a slightly smaller effect of correlation~none for xxx!than we do, and in the opposite direction for the average value. All in all correlation plays a minor role for the mag- netizability of HCN. Notice that we could not perform cor- related calculations with basis set IVb and CAS B.

Correlation plays a greater role for the magnetizability polarizabilities. In one case (hzz,xx) the effect is quite dra- matic but, as for the two previous molecules, the influence on the anisotropyDh is quite small~less than 3%!. In axial molecules one obtains from Eq.~41!

Dh5 15¹ ~7hxx,xx25hxx,y y12hzz,zz22hxx,zz

22hzz,xx112hxz,xz! ~43!

which shows that the contribution ofhzz,xx ~219.00 a.u.!to the anisotropy is, for instance, about an order of magnitude smaller than that of hxx,y y ~2100.37 a.u.!, which is only slightly influenced by correlation.

TABLE II. C2H2magnetizabilitiesxab, hypermagnetizabilitieshab,cdand anisotropyDh~a.u.!. See the text for details.

SCF Basis IVb

Correlated Basis IVa-CAS B

xxx 24.71^a 24.65^b

xzz 25.21^a 25.06^b

hxx,xx 277.42 275.16

hxx,y y 2205.34 2202.72

hxx,zz 2193.08 2182.24

hzz,xx 284.24 289.68

hzz,zz 285.58 286.36

hxz,xz 38.87 37.01

Dh^{c 88.98 86.85}

aReference values:24.748 (xxx),25.223 (xzz), Ref. 47~IGLO!.

bReference values: 25.339 (xxx),25.074 (xzz), Ref. 41~MCSCF, gauge origin in the center of mass!.

cExperiment: 20654~Ref. 57. Notice the use of Ko¨nig’s definition forDh^, see the text!and 455634~Ref. 31!.

TABLE III. HCN magnetizabilities xab, hypermagnetizabilitiesjab,cand hab,cdand anisotropyDh~a.u.!. The sequence of the indices ab – c injab,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

xxx 23.10^a 23.20^b

xzz 24.45^a 24.35^b

jxx,z 5.08 4.88

jzz,z 0.06 0.38

jxz,x 0.90 1.08

hxx,xx 229.90 233.44

hxx, y y 298.82 2100.37

hxx,zz 279.40 268.75

hzz,xx 22.38 219.00

hzz,zz 242.38 241.67

hxz,xz 22.86 21.75

Dh ^{42.53 41.40}

aReference values:23.118 (xxx),24.447 (xzz), Ref. 45, CHF.

bReference values:23.118 (xxx),24.370 (xzz), Ref. 45, MP2.