Gauge-origin independent magneto-optical activity within coupled cluster response theory

Gauge-origin independent magneto-optical activity within coupled cluster response theory

Sonia Coriani,^a)Christof Ha¨ttig,^b)and Poul Jørgensen

Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 A˚ rhus C, Denmark Trygve Helgaker

Department of Chemistry, University of Oslo, P. O. B. 1033, N-0315 Oslo, Norway 共Received 2 May 2000; accepted 9 June 2000兲

A gauge-origin invariant formulation of the frequency-dependent Verdet constant V(␻^{) of} magneto-optical rotation and of the Faraday B term of magnetic circular dichroism for coupled-cluster wave functions is derived within the framework of variational response theory.

Working expressions suitable for implementation in ab initio program packages are presented.

These expressions have a structure similar to that of the expressions for the first hyperpolarizability and the two-photon transition moment, respectively, for the Verdet constant and theB term. The approach is general and can easily be extended to other similar frequency-dependent properties as well as to other wavefunction models. Pilot results at the CCSD level are presented for V(␻^{) of HF} and H₂. © 2000 American Institute of Physics.关S0021-9606共00兲30233-1兴

I. INTRODUCTION

The discovery of magneto-optical activity共MOA兲dates back to 1845, when Faraday^1,2observed that the application of a magnetic field in the direction of propagation of a plane polarized light beam passing through a piece of heavy glass affected the behavior of the beam. The effect consisted of a rotation共␾兲of the plane of polarization of the ray induced by the longitudinal magnetic field; it was later found to occur in any sample and shown to be proportional to the magnetic field strength. The proportionality constant—which is fre- quency and temperature dependent—is known as the Verdet constant V(␻^).3When the frequency approaches the absorp- tion region of the sample, the light is attenuated and the rotation is accompanied by the development of an ellipticity 共␪兲. Both aspects, rotation and ellipticity, are related to a differential behavior induced by the magnetic field of the right 共R兲and left共L兲circularly polarized components of the plane polarized light beam. Traditionally, the rotation is as- sociated with the anisotropy of the refractive indices n_L⫺n_R and the ellipticity with the anisotropy of the absorp- tion coefficients n_L⬘⫺n_R⬘. The latter effect is also called mag- netic circular dichroism 共MCD兲. Magneto-optical rotation 共MOR兲dispersion and MCD are also referred to as the ‘‘Far- aday effect.’’

The fundamental equations relating the observed rotation and/or ellipticity to the microscopic properties of the sample have been derived in several equivalent ways. Buckingham and Stephens⁴ used a conventional refractive-index ap- proach, relating the complex rotation to the anisotropy of the complex refractive indices for the right and left circularly polarized components. The latter are expressed through the

Maxwell equations in terms of averaged complex induced oscillating molecular moments, which are calculated quantum-mechanically. Focusing on the MCD, Stephens^5–7 rederived the equations for the ellipticity by means of semi- classical radiation absorption theory. Equivalent equations for the rotation and ellipticity have also been obtained by the refringent scattering approach, as shown for example by Barron.⁸ Both the conventional refractive-index approach and the refringent-scattering approach have been success- fully applied to several optical effects such as natural optical activity, linear birefringence, and Raman optical activity.

As for other optical properties, electron correlation is expected to play an important role in the ab initio calculation of the molecular properties characterizing the Faraday effect—that is, the Verdet constants and the Faraday A,B, andC terms. In addition, the problem of the unphysical de- pendence on the origin of the vector potential共gauge-origin兲 for magnetic properties calculated with conventional finite basis sets needs to be addressed.

In the past, the calculation of molecular Verdet constants has appeared at various levels of ab initio theory 共see, for example, Refs. 9–13兲. The Verdet constant has been calcu- lated either from a spectral-representation expression,¹³or as combination of quadratic dipole–dipole–magnetic dipole re- sponse functions.^9–12 Using the Becquerel approximation,¹⁴ which is exact only in the atomic case, propagators or linear response function expressions have also been used.^9,15These calculations have mostly been limited to systems that are gauge-origin independent by symmetry; no general solution has so far been presented for a gauge-origin invariant calcu- lation of Verdet constants. Indeed, in a recent paper, Peder- sen et al. question the possibility of obtaining gauge-origin independent Verdet constants using a conventional coupled cluster ansatz for the wave function.¹⁶

Theoretical calculations of the Faraday terms of the MCD—in particular the B term—have mainly been per-

a兲Permanent address: Dipartimento di Scienze Chimiche, Universita` degli Studi di Trieste, Via Giorgieri 1, I-34127 Trieste, Italy.

b兲Present address: Forschungszentrum Karlsruhe, Institute of Nanotechnol- ogy, P.O. Box 3640, D-76021 Karlsruhe, Germany.

3561

0021-9606/2000/113(9)/3561/12/$17.00 © 2000 American Institute of Physics

formed by means of sum-over-states procedures. Recently, we presented an investigation of the MCD of two naturally gauge-invariant systems, reformulating theBterm as a single residue of the dipole–dipole–magnetic dipole quadratic re- sponse function in a SCF and MCSCF parameterization.¹⁷A gauge-invariant method for calculating the Bterm was pre- sented in the early 1970s by Seaman and Linderberg.¹⁸Self- consistent molecular orbitals at nonzero magnetic field were determined in the Pariser–Parr–Pople model, starting with an atomic orbital basis of London gauge including orbitals 共LAOs or GIAOs兲,^19–21 and then employed to calculate transition moments according to the single-excited configuration-interaction and the random-phase approxima- tions. TheBterm was then obtained by extrapolating to zero magnetic field the ratio between the field-dependent transi- tion strength and the magnetic field strength—that is, essen- tially by numerical differentiation.

Inspired by Seaman and Linderberg’s approach, we present here a gauge-origin invariant formulation of magneto-optical activity within the framework of coupled cluster共CC兲response theory.^22–26Our approach relies on the use of LAOs in connection with a reformulation of both the Verdet constant and the FaradayB term as total derivatives with respect to the strength of an external static magnetic field of, respectively, the dipole polarizability and dipole transition strength. For the latter properties, we employ the appropriate CC response expressions defined with respect to a SCF wave function optimized within the magnetic field.

For the Verdet constant, this represents the first attempt to extend the London-orbital approach to the calculation of a frequency-dependent quadratic response property. To illus- trate the validity of the proposed approach pilot results for the Verdet constant of HF at the CCSD level are finally presented. Also, the effect of using LAOs when calculating Verdet constants which are gauge-origin independent by symmetry is tested through preliminary calculations on H₂. II. THE BASIC EQUATIONS

A. Conventional formulation

To treat MOR and MCD in a unified way, one may in analogy with Ref. 4 define the complex optical rotation␾^{˜ per} unit of path length

␾

˜⫽␾⫹i␪⫽ ␻

2c⌬˜ ,n 共1兲

where ⌬˜n⫽(n_L⫺n_R)⫹i(n_L⬘⫺n_R⬘) is the anisotropy of the complex refractive index. As shown for example in Refs. 4 and 8, the complex optical rotation for plane-polarized light traveling in the z direction of a space-fixed frame is related to the complex polarizability ␣^˜_␣␤ 共frequency argument im- plied兲,

␾˜⫽¹4␻␮0cN关i具␣^˜xy⫺␣^˜y x典兴, 共2兲 where具¯典indicates an appropriate statistical average. In the nonabsorptive region, the complex polarizability ␣^˜␣␤ is de- fined as

␣

˜_␣␤⫽2

ប

再

^j

^兺

^{⫽n ␻jn2␻⫺jn␻2R共具n兩d␣兩j典具j兩d␤兩n典兲}

⫹i

兺

_{j⫽n ␻jn}²_⫺^␻_␻^{2I共具n兩d␣兩j典具j兩d␤兩n典兲}

冎

⫽␣_␣␤⫺i␣_␣␤⬘ ⫽␣^˜_␤␣* , 共3兲 where d indicates the electric dipole moment operator and Eq. 共2兲reduces to

␾⫽⫺¹2␻␮0cNI具␣^˜xy典⫽¹2␻␮0cN具␣xy⬘ 典^. 共4兲 These definitions are readily generalized to the absorptive regions by introducing the line-shape functions f and g共see, for example, Ref. 8兲. For simplicity, we here restrict our- selves to the nonabsorptive region.

A linear dependence on the magnetic field is brought explicitly into Eq. 共4兲through the linear dependence of the antisymmetric polarizability ␣xy⬘ on the magnetic field and the orientational effect of the magnetic field on the perma- nent molecular magnetic moments. The polarizability␣_␣␤⬘ ^is expanded in the magnetic field according to

␣_␣␤⬘ 共B兲⫽␣_␣␤⬘ 共0兲⫹␣_␣␤⬘^共m,␥^兲B_␥⫹O共B²兲, 共5兲 where B represents the magnetic field strength. Implicit sum- mation over repeated indices is assumed here and through- out. The sum-over-states expression for the higher-order po- larizability ␣_␣␤⬘^(m),␥ is obtained from the definition of ␣_␣␤⬘ ^by substituting the magnetic field-dependent eigenstates j ,n and their frequency separations␻jn⫽␻j⫺␻n by the correspond- ing perturbational expansions in terms of zero-field eigen- states and frequency separations,^8,27

␣_␣␤,␥⬘^共m兲⫽⫺ 2

ប²_j

兺

_⫽n

再

^{共␻22jn␻␻⫺␻jn2兲2共m␥j⫺m␥n兲I共具n兩d␣兩j典具j兩d␤兩n典兲⫹共␻2jn␻⫺␻2兲I}

冋

^k

^兺

^{⫽n 具k兩␻mkn␥兩j典}

^共

^{具n兩d␣兩j典具j兩d␤兩k典}

⫺具ⁿ兩d_␤兩j典具^j兩d_␣兩k典

兲

^⫹_k

兺

_⫽j ^具j兩_␻^m_{k j}^␥兩k典

^共

^{具n兩d␣兩j典具k兩d␤兩n典⫺具n兩d␤兩j典具k兩d␣兩n典}

^兲 册冎

⫽⫺2

ប_j

兺

_⫽n

再

^{ប共␻22jn␻␻⫺jn␻2兲2A␣␤␥⬘共jn兲⫹共␻jn2␻⫺␻2兲B␣␤␥⬘共jn兲}

冎

^{. 共6兲}

Here m_␥ is a Cartesian component of the magnetic dipole operator and m_␥^k⫽具^k兩m␥兩k典. The states n, j,k and the frequency separations ␻kn, ␻k j, ␻jn refer now to the unpertubed system (B⫽0). The orientational effect on the magnetic moment is

accounted for either by means of classical weighted Boltz- mann average with the potential energy U

⫽⫺m_␥ⁿB_␥or by means of an unweighted quantum statistical average.⁸ For a fluid sample, the resulting rotation with re- spect to the molecular frame becomes

␾⫽ 1

12␻␮0cNB_z⑀_␣␤␥

冉

^{␣␣␤,␥⬘共m兲⫹kT1 m␥n␣␣␤⬘}

冊

^{⫽V共␻兲Bz,}

共7兲 where we have introduced the Verdet constant V(␻⁾

⫽␾^/Bz. The symbol ⑀␣␤␥ was used for the Levi–Civita tensor. Note that, if n is the ground state, the second term in parentheses is zero for closed-shell systems as the magnetic dipole is quenched. An analogous expansion for ␣^˜_␣␤ ^fol- lowed by orientational averaging leads to a similar equation for the generalized complex rotation Eq.共2兲.

The 共averaged兲Faraday A,B, and Cterms, used to ra- tionalize the MCD, are easily identified from the tensors so far introduced by splitting the total rotation 共or more rigor- ously the complex rotation兲 into contributions from a par- ticular ‘‘transition’’ n→j —that is, ␾⫽兺j⫽n␾⁽ⁿ→j ). Ac- cording to the current convention,^8,28

␾共n→j兲⫽⫺␮0cNB_z

3ប

再

^{ប共␻2␻2jn2⫺␻␻jn2兲2A共n→j兲}

⫹ ␻²

␻jn

2 ⫺␻²

冋

^{B共n→j兲⫹kT1 C共n→j兲}

册 冎

^{, 共8兲}

A共n→j兲⫽¹6⑀_␣␤␥共3A_␣␤␥⬘^共jn兲兲

⫽¹2⑀_␣␤␥共m_␥^j⫺m_␥ⁿ兲I共具ⁿ兩d_␣兩j典具^j兩d_␤兩n典兲, 共9兲 B共n→j兲⫽¹6⑀_␣␤␥共3B_␣␤␥⬘^共jn兲兲

⫽⑀_␣␤␥I

冋

^k

^兺

^{⫽n 具kប兩m␻␥kn兩n典具n兩d␣兩j典具j兩d␤兩k典}

⫹k

兺

⫽j

具^j兩m_␥兩k典

ប␻k j 具ⁿ兩d_␣兩j典具^k兩d_␤兩n典

册

^{, 共10兲}

C共n→j兲⫽¹2⑀␣␤␥m_␥ⁿI共具ⁿ兩d_␣兩j典具^j兩d_␤兩n典兲. 共11兲 For molecules with a nondegenerate ground state n and a nondegenerate state j only theBterm is nonvanishing. In the following, we are concerned with the calculation of B(n→j ) and of the Verdet constant V(␻^).

B. Derivative formulation

The Verdet constant and the FaradayBterm may alter- natively be expressed as derivatives of property expressions with respect to the magnetic induction. Expanding the rota- tion in the field according to²⁹

具␾典B⫽具␾典⫹B_z

再 冓 ^冉

^dBd␾z

^冊

⁰

冔

^{⫹kT1 共具␾mzn典⫹具␾典具mzn典兲}

冎

⫹O共B²兲

⫽B_z

再 冓 ^冉

^dBd␾z

^冊

⁰

冔

^{⫹kT1 具␾mzn典}

冎

^{⫹O共B2兲, 共12兲}

where the averages on the r.h.s. are taken at zero magnetic field, we can immediately relate the Verdet constant for a closed-shell system to the derivative of the frequency- dependent antisymmetric polarizability 共i.e., the imaginary part of the complex polarizability兲,

V共␻兲⫽

冓 ^冉

^dBd␾z

^冊

⁰

冔

⫽ 1

12␻␮0cN⑀_␣␤␥

冉

^{d␣␣␤⬘ dB共⫺}␥^␻;␻兲

冊

0

⫽⫺ 1

12␻␮0cN⑀_␣␤␥I

冉

^{d␣˜␣␤dB共⫺}␥^␻;␻兲

冊

0

共13兲 and identify the FaradayB(n→j) term of a closed-shell sys- tem without degenerate states from the derivative of a one- photon transition strength

B共n→j兲⫽1

2⑀␣␤␥

冉

^{dI共具n兩d␣dB兩j典具}␥^{j兩d␤兩n典兲}

冊

0

⫽1

2⑀␣␤␥I

冉

^{dSdBn j␣␤}␥

冊

0

. 共14兲

The derivative formulation is advantageous for several reasons. First, contrary to the conventional sum-over-states representation, it does not require that all excited states are explicitly calculated. Rather, one can simply identify the property expressions to be differentiated in terms of linear response functions and corresponding residues, and then dif- ferentiate these response function expressions for a given wave function approximation to obtain the Verdet constant and the Faraday B term. Second, we also demonstrate that, with London atomic orbitals and using response function ex- pressions where the orbitals are allowed to relax in the mag- netic field, V(␻^{) and}Bmay be obtained gauge-origin inde- pendent without consideration whether the Ehrenfest theorem is satisfied for one-electron operators in the chosen wavefunction approximation. Difficulties in ensuring gauge- origin independence are encountered if V(␻^{) and}Bare cal- culated in terms of quadratic response functions and residues of quadratic response functions.¹⁰

III. GAUGE-ORIGIN INVARIANT MOA WITHIN CC RESPONSE THEORY

As physical observables, the Verdet constant V(␻^{) and} the FaradayBterm must be independent of the choice of the gauge origin. In approximate calculations, however, gauge- origin independence is not necessarily satisfied. In the next subsections, we describe in detail how gauge-origin indepen- dent expressions for V(␻^{) and} Bcan be obtained from the previous derivative formulation of the response properties and the use of London orbitals. We restrict ourselves to the CC model but emphasize that the same strategy may be ap- plied for other approximate wave function models.

First we summarize how the polarizability and oscillator strength can be calculated in CC response theory. We then discuss the use of LAOs to parametrize the magnetic field dependence in the polarizability and oscillator strength. In

particular, we demonstrate that the magnetic-field orbital- relaxed property expressions to be differentiated are indepen- dent of the chosen connection scheme and identical to that obtained using the symmetric connection. As the molecular orbital integrals in the symmetric connection³⁰ are indepen- dent of the gauge origin, we obtain a gauge-invariant Verdet constant and Bterm.

A. The ansatz

In CC response theory, the frequency-dependent dipole polarizability and the dipole oscillator strengths can be for- mulated as derivatives with respect to the 共frequency- dependent兲 electric-field strengths of the real part of the time-averaged quasienergy Lagrangian^26,31 and the time- averaged transition moment Lagrangians,³²both of which are generalizations of the energy Lagrangian introduced for the treatment of time-independent properties.³³ To obtain these derivatives, a time-dependent CC state is used in which the orbitals are not allowed to relax with respect to the electric perturbation.^26,31,32 The time-dependent Hamiltonian em- ployed can be written as

H共t,E兲⫽H⫹V共t,E兲

⫽H⫺d_␥关E_␥共␻␥兲exp共⫺i␻␥t兲

⫹E_␥共⫺␻␥兲exp共i␻␥t兲兴 共␥⫽x,y ,z兲, 共15兲 where H is the Hamiltonian for the unperturbed system, E(␻) the electric field strength, and d the static electric di- pole moment operator.

The resulting expressions for the electric-field-unrelaxed frequency dependent polarizability^26,31 and for the one- photon transition strength^26,32are

␣_␣␤共␻_␣^,␻_␤兲⫽⫺¹2Cˆ⫾␻兵␩^d␣td␤共␻_␤兲⫹Ft^d␣共␻_␣兲t^d␤共␻_␤兲

⫹␩^d␤td␣共␻␣兲其 ^共16兲 with␻␣⫽⫺␻␤⫽␻^{, and}

S_{n j}^␣␤⫽¹2兵^Mn←j d_␣

M^d_j←^␤_n⫹共M_n^d_←^␤ _jM_j^d_←^␣_n兲*其^{, 共17兲} where the left and right one-photon dipole transition mo- ments M_n^d_←jand M^d_j←n are given by^26,32

M_n^d_←j⫽^¯␰^d共⫺␻j兲E^j共␻j兲⫽␩^dEj共␻j兲⫹M¯ ^j共␻j兲␰^d, 共18兲 M^d_j←n⫽E¯^j共⫺␻j兲␰^d. 共19兲 The operator Cˆ⫾␻ enforces time-reversal symmetry on the CC polarizability by symmetrizing with respect to simulta- neous complex conjugation and inversion of the sign of the frequency. The vector-matrix notation of Refs. 26 and 32 was used. The various matrices and vectors are defined as^26,31,32

␰_␮^d⫽具␮^¯兩d兩CC典^{, 共20兲}

␩_␮^d⫽具^⌳兩关d,␶␮兴兩CC典^{, 共21兲} F_␮␯⫽具^{⌳兩关关H,}␶␮兴,␶_␯兴兩CC典^{, 共22兲} and

¯␰^d共␻d兲⫽␩^d⫹Ft^d共␻d兲, 共23兲

where

兩CC典^⫽exp共T兲兩HF典^{, 共24兲}

具␮^¯兩⫽具^HF兩␶_␮^{† exp}共⫺T兲⫽具␮兩exp共⫺T兲, 共25兲 具^⌳兩⫽具^HF兩

冉

^1⫹

^兺

^{␮ t¯␮␶␮†}

冊

^{exp共⫺T兲. 共26兲}

The cluster operator is given by T⫽兺_␮t_␮␶_␮^{, with} ␶_␮ ^an excitation operator and t_␮ the associated amplitude共for each excitation level ␮兲. To zero order with respect to the time- dependent electric-field perturbation, the amplitudes t are ob- tained from the CC equations,

具␮兩exp共⫺T兲H exp共T兲兩HF典⫽0. 共27兲 The zero-order Lagrange multipliers t¯ are the solutions of the set of linear equations

␩⫹t¯A⫽0, 共28兲

where

␩␮⫽具^HF兩关H,␶␮兴兩CC典^, 共29兲 and the CC Jacobian A is given by

A_␮␯⫽具␮^¯兩关H,␶_␯兴兩CC典^. 共30兲 The left and right excitation vectors E¯^j(⫺␻j) and E^j(␻j) are the solutions of the linear equations

E¯^jA⫽␻jE¯^j, AE^j⫽␻jE^j 共31兲 under the biorthonormality condition E¯^jE^f⫽␦j f. The fre- quency argument in E¯^jand E^jis conventional.^32,34The first- order responses t^d(␻d) of the amplitudes to the electric field are obtained from the equations

␰^d⫹关A⫺␻d1兴t^d共␻d兲⫽0. 共32兲 The intermediate vector M¯ ^j(␻j) is found by solving³²

M¯ ^j共␻j兲共A⫹␻j1兲⫽⫺FE^j共␻j兲. 共33兲 In second quantization, the dipole and Hamiltonian operators are expressed as

d⫽

兺

_pq ^{dpqEpq, 共34兲}

H⫽

兺

_pq ^hpqEpq⫹12_pqrs

兺

^{gpqrsepqrs, 共35兲}

where E_pq⫽兺_␴⫽␣,␤a_p^†_␴a_q␴, e_pqrs⫽E_pqE_rs⫺␦qrE_ps, and d_pq, h_pq and g_pqrs are the one- and two-electron integrals over the orthonormal basis of molecular orbitals.

B. The introduction of the magnetic field

In the presence of an external static magnetic field, the polarizability and the oscillator strength may be obtained in a two-step approach, replacing the unperturbed Hamiltonian in Eq.共15兲by the Hamiltonian for the system in presence of the static magnetic field

H共B兲⫽H⫺m_O,␥B_␥⫺ ¹2␰O,␥␦B_␥B_␦ 共␥^,␦⫽x,y ,z兲 共36兲

and treating all states as magnetic-field dependent. The time- dependent Schro¨dinger equation for the interaction between the polarized light and the system is thus solved for a static Hamiltonian that contains the interaction with the magnetic field.

In the Hamiltonian, both the magnetic dipole moment operator m_O and the diamagnetic magnetizability operator

␰O refer to an arbitrarily chosen gauge origin O. Alterna- tively, the Hamiltonian H(B) can be written in terms of the vector potentials of the electrons with respect to the gauge origin A_i⬅A(r_iO)⫽¹2B⫻(r_i⫺O) as

H共A兲⫽1

2

兺

_i ^{共⫺iⵜi⫹Ai兲2⫺}

兺

_i,K r^Z_iK^K⫹1

2

兺

_i⫽j r¹_{i j}. 共37兲 The dependence of the Hamiltonian H(B) on an arbitrarily chosen gauge origin is a manifestation of the gauge-origin problem, which hampers the calculation of magnetic molecu- lar properties within approximate methods.

Thus, in the presence of a magnetic field, the polarizabil- ity and the oscillator strength are still defined according to Eqs. 共16兲–共33兲, where all contributions become magnetic- field dependent, for example, the Jacobian matrix becomes

A_␮␯共B兲⫽具␮^¯共B兲兩关H共B兲,␶_␯共B兲兴兩CC共B兲典^{, 共38兲} where 兩CC(B)典^{⫽exp T(B)兩HF(B)}典 and analogously for 具␮¯ (B)兩.

In terms of an orthonormal set of field-dependent orbit- als, the second-quantization form of the Hamiltonian H(B) becomes^30,35

H共A兲⫽

兺

_pq ^{hpq共A兲Epq共A兲⫹1}2_pqrs

兺

^{gpqrs共A兲epqrs共A兲,}

共39兲 where the field dependence in the one-electron part arises from the implicit field dependence in the orbitals and the creation/annihilation operators as well as from the explicit dependence in the one-electron operator itself. Note that, in the second quantization formalism, also the dipole moment operator becomes magnetic field dependent, through the im- plicit dependence in the magnetic field of the orbitals and of the creation/annihilation operators.

C. The London orbitals and the relationship between different connections

Suppose that we have performed a Hartree–Fock calcu- lation at zero magnetic field and obtained the wave function 兩HF典and a set of optimized orbitals

␺p共0兲⫽

兺

_␮ ^{␹␮C共␮0兲p, 共40兲}

where ␹␮ is a conventional Gaussian orbital centered on nucleus M at position R_M. For any given magnetic induction B, a basis of atomic field-dependent orbitals, the London orbitals,¹⁹兵␸␮其 can be defined according to

␸␮共B兲⫽exp共⫺iA_{M O}^e •r兲␹␮, 共41兲

where A^e_{M O} is the vector potential at the position R_{M O}⫽R_M⫺O of the nucleus M with respect to the chosen gauge origin,

A_MO^e ⫽¹2B⫻R_{M O}, 共42兲 and r is the electron coordinate. Note that in absence of the perturbation ␸␮(0)⫽␹␮.

The LAOs can be employed to define an orthonormal- ized basis of molecular field-dependent orbitals, from which an Hamiltonian operator defined at all values of the field can be constructed. This may be accomplished in two steps.

First, from the LAOs and the MO coefficients C_␮n⁽⁰⁾ of the optimized orbitals at zero magnetic field, the unmodified mo- lecular orbitals共UMOs兲are constructed as³⁰

␾q共B兲⫽

兺

_␮ ^{␸␮共B兲C␮共0兲q. 共43兲}

Second, assuming that their overlap matrix S is nonsingular, the UMOs are further transformed into a set of orthonormal- ized molecular orbitals 共OMOs兲,³⁰

␺p共B兲⫽

兺

_q ^{␾q共B兲Rq p共B兲. 共44兲}

The transformation matrix R⫽S^⫺1/2U, where U is unitary defines the different connection schemes^36,37which establish a one-to-one correspondence between orbitals at different values of the field. If U⫽1, the OMO basis in Eq. 共44兲cor- responds to a symmetric orthogonalization of the UMO basis—the symmetric connection.³⁰For any general connec- tion scheme,

␺p

g⫽

兺

_r ^{␺rsUr pg ⫽}

兺

_r ^{␺rs关e⫺xg兴r p, 共45兲}

where the unitary matrix U^g was expressed in terms of an anti-Hermitian matrix x^g. The indices ‘‘g’’ and ‘‘s’’ are used to distinguish between a general connection and the symmet- ric connection and explicit reference to the field dependence is suppressed for ease of notation. The orbital indices p, q, r, s are used for unspecified 共occupied or virtual兲 orbitals, i, j, k, l for occupied, and a, b, c, d for vir- tual. Note that the OMOs are not the optimized orbitals in the presence of the external perturbation but that they be- come identical to the optimized orbitals for the unperturbed system in absence of the perturbation.

D. The„effective…relaxed Hamiltonian and dipole operator

We now consider a general connection OMO basis and employ it to expand the Hamiltonian and parametrize the optimized Hartree–Fock state in the presence of the field according to共neglecting the purely nuclear contributions兲

H共B兲⫽

兺

_pq ^{hpqg Epqg ⫹1}2_pqrs

兺

^{gpqrsg epqrsg , 共46兲}

兩HF共B兲典⫽exp共⫺␬^˜g兲兩HF_g典^, 共47兲

where兩HF_g典 is a determinantal wave function built from the creation operators for the general OMO basis, and the opera- tor

␬

˜_g⫽

兺

pq nonred.

␬

˜^g_pqE_pq^g 共48兲

generates共nonredundant兲orbital rotations among the OMOs.

Note that E_pq^g ,兩HF_g典^{, and}␬^˜pq

g are magnetic field-dependent, but their dependence is suppressed for ease of notation.

To obtain V(␻^{) and}Bwe need to take the total deriva- tive of matrix elements such as Eq.共38兲, that is, the deriva- tive of generalized transition expectation values between states defined at the same value of the field. As a conse- quence of Wick’s theorem, we can then neglect the magnetic field 共and gauge兲 dependence of the elementary creation/

annihilation operators corresponding to the OMO basis.^30,33 As discussed in Ref. 37, this may be viewed as expanding the elementary operators at a given value of the field in the elementary operators at zero field. The Hamiltonian in Eq.

共46兲and the field-dependent Hartree–Fock state in Eq. 共47兲 can then be written in the ‘‘effective’’ form,

H共B兲⫽H_g⫽

兺

pq

h^g_pqE_pq⫹1 2pqrs

兺

g^g_pqrse_pqrs, 共49兲

兩HF共B兲典⫽exp共⫺␬g兲兩HF典^, 共50兲

where E_pq, e_pqrs, and 兩HF典 refer to the optimized basis in absence of the perturbation, and

␬g⫽^nonred.

兺

_pq ^{␬gpqEpq. 共51兲}

The elements␬pq

g of the anti-Hermitian matrix␬^{g, which} depend implicitly on the field, are nonzero only for the occupied-virtual and virtual-occupied blocks—that is, they are the nonredundant parameters as determined from the共ef- fective兲Hartree–Fock orbital optimization condition, 具^HF共B兲兩关E_pq共B兲,H共B兲兴兩HF共B兲典^⫽0

→具^HFg兩exp共␬g兲关exp共⫺␬g兲E_pqexp共␬g兲,H_g兴

⫻exp共⫺␬g兲兩HF_g典⫽具^HF兩关E_pq,H_g^␬兴兩HF典⫽0, 共52兲 where we have introduced the 共effective兲 relaxed Hamil- tonian in the given general-connection OMO basis

H_g^␬⫽exp共␬g兲H_gexp共⫺␬g兲. 共53兲 In the following, we shall consider the structure of this op- erator.

Inserting Eq. 共45兲into the definition of the integrals of H_g and introducing the anti-Hermitian operator,

x_g⫽

兺

pq

x^g_pqE_pq, 共54兲

where x_pq^g are the elements of the anti-Hermitian matrix x^g, the Hamiltonian H_g may be written as

H_g⫽exp共x_g兲H_sexp共⫺x_g兲, 共55兲 where

H_s⫽

兺

_pq ^{hpqs Epq⫹1}2_pqrs

兺

^{gspqrsepqrs 共56兲}

is the effective Hamiltonian in the symmetric connection.

With this reformulation of H_g, the relaxed Hamiltonian in the given general-connection OMO basis becomes

H_g^␬⫽exp共␬g兲exp共x_g兲H_sexp共⫺x_g兲exp共⫺␬g兲

⫽exp共␬rd兲exp共␬s兲H_sexp共⫺␬s兲exp共⫺␬rd兲

⬅exp共␬rd兲H_s^␬exp共⫺␬rd兲, 共57兲 where

␬s⫽

兺

pq nonred.

␬pq

s E_pq, 共58兲

␬rd⫽

兺

^occ_{i j} ^{␬i jrdEi j⫹}

兺

_ab^{vir ␬abrdEab, 共59兲}

with the anti-Hermitian matrices ␬^sand␬^rdhaving nonzero elements corresponding to the nonredundant and redundant orbital rotations, respectively. The equivalence of the second and third expressions in Eq.共57兲follows since these expres- sions represent two equivalent parameterizations of a general unitary matrix.³⁸

Let us derive a set of equations for the nonredundant parameters␬^s. Inserting Eq. 共57兲in Eq.共52兲, we obtain 具^HF兩关E_pq,exp共␬rd兲H_s^␬exp共⫺␬rd兲兴兩HF典

⫽具HF兩exp共␬rd兲关exp共⫺␬rd兲E_pqexp共␬rd兲,H_s^␬兴

⫻exp共⫺␬rd兲兩HF典⫽0, 共60兲 where pq refers to the nonredundant parameters ( pq⫽ai, ia). Because of the relations

exp共⫺␬rd兲兩HF典⫽兩HF典^{, 共61兲}

exp共⫺␬rd兲E_pqexp共␬rd兲⫽^nonred.

兺

_rs ^{CrsErs 共62兲}

we find that Eq.共52兲 关or equivalently Eq.共60兲兴is satisfied if 具^HF兩关E_rs,exp共⫺␬s兲H_sexp共␬s兲兴兩HF典^{⫽0. 共63兲} This corresponds to the effective Hartree–Fock optimization condition in the symmetric connection, from which the non- redundant parameters, and thus the effective relaxed H_s^␬ op- erator, are determined.

The redundant parameters␬^{rdof Eq.}共57兲are not deter- mined by the orbital optimization condition, but are implic- itly given through the choice of ␬g, x_g, and ␬s. Since the relaxed Hamiltonian H_g^␬is later used only in contexts where the property considered is invariant to redundant orbital ro- tations, explicit knowledge of these parameters is therefore not required.

Thus, even though the parameters␬^gand␬^sdetermined by Eq.共52兲and Eq.共63兲, respectively, are not the same, the final effective relaxed Hamiltonian in the general connection is for all practical purposes 共i.e., to within a redundant or- bital transformation兲 equivalent to the effective relaxed Hamiltonian in the symmetric connection. This is an impor-

tant result also in the sense that the symmetric connection may not always be the most useful approach from a numeri- cal point of view.³⁹ Similar relations may be derived for other operators. In particular, for the effective relaxed dipole moment operator, we obtain

d_g^␬⫽exp共␬g兲d_gexp共⫺␬g兲

⫽exp共␬g兲exp共x_g兲d_sexp共⫺x_g兲exp共⫺␬g兲

⫽exp共␬rd兲exp共␬s兲d_sexp共⫺␬s兲exp共⫺␬rd兲

⬅exp共␬rd兲d_s^␬exp共⫺␬rd兲, 共64兲 with d_g⫽兺pqd_pq^g E_pq and d_s⫽兺pqd_pq^s E_pq.

E. The effective CC equations and matrix elements In the presence of the magnetic field, the CC wave func- tion is parametrized in the OMO basis as

兩CC共B兲典^⫽exp共⫺␬^˜g兲exp共˜T

g兲兩HF_g典^{, 共65兲}

where T˜

g⫽兺_␮t˜_␮^g␶_␮^g. Analogously, 具␮^¯共B兲兩⫽具^HFg兩␶_␮^g†exp共⫺T˜

g兲exp共^˜␬g兲

⫽具␮g兩exp共⫺T˜

g兲exp共␬^˜g兲, 共66兲

具⌳共B兲兩⫽具^HFg兩

冉

^1⫹

^兺

^{␮ tន␮g␶␮g†}

冊

^{兩exp共⫺T˜g兲exp共␬˜g兲. 共67兲}

Again, according to Wick’s theorem, we can expand our creation/annihilation OMO basis in the generator basis at the expansion point, only retaining an implicit dependence on the magnetic field in the orbital and wavefunction param- eters,

兩CC共B兲典⫽exp共⫺␬g兲exp共T_g兲兩HF典^, 共68兲

具␮^¯共B兲兩⫽具^HF兩␶_␮^{† exp}共⫺T_g兲exp共␬g兲

⫽具␮兩exp共⫺T_g兲exp共␬g兲, 共69兲 具⌳共B兲兩⫽具^HF兩

冉

^1⫹

^兺

^{␮ t¯␮g␶␮†}

冊

^{兩exp共⫺Tg兲exp共␬g兲, 共70兲}

where T_g⫽兺_␮t_␮^g␶_␮^.

The CC amplitudes are obtained by solving the 共effec- tive兲relaxed CC equations,

具␮兩exp共⫺T_g兲H_g^␬exp共T_g兲兩HF典⫽0. 共71兲 Introducing the relation between the operators in general and symmetric connection, Eq.共57兲, we get

具␮兩exp共⫺T_g兲H_g^␬exp共T_g兲兩HF典

⫽具␮兩exp共⫺T_g兲exp共␬rd兲H_s^␬exp共⫺␬rd兲exp共T_g兲兩HF典

⫽0⬅具␮兩exp共␬rd兲exp共⫺T⬘兲H_s^␬exp共T⬘兲兩HF典^{⫽0, 共72兲} and the effective expressions for the vectors and matrix ele- ments 共for the general connection兲 to be differentiated be- come

␰_␮^d⫽具␮兩exp共⫺T_g兲d_g^␬exp共T_g兲兩HF典

⫽具␮兩exp共⫺T_g兲exp共␬rd兲d_s^␬exp共⫺␬rd兲exp共T_g兲兩HF典

⫽具␮兩exp共␬rd兲exp共⫺T⬘兲d_s^␬exp共T⬘兲兩HF典^, 共73兲

␩_␮^d⫽具^HF兩

冉

^1⫹

^兺

^{␭ t¯␭g␶␭†}

冊

^{exp共⫺Tg兲关dg␬,␶␮兴exp共Tg兲兩HF典}

⫽具^HF兩

冉

^1⫹

^兺

^{␭ t¯␭g␶␭†}

冊

^{exp共⫺Tg兲}

⫻关exp共␬rd兲d_s^␬exp共⫺␬rd兲,␶_␮兴exp共T_g兲兩HF典

⫽具^HF兩

冉

^1⫹

^兺

^{␭ t¯␭g␶␭†}

冊

^{exp共␬rd兲exp共⫺T⬘兲}

⫻关d_s^␬, exp共⫺␬rd兲␶_␮^exp共␬rd兲兴exp共T⬘兲兩HF典^, 共74兲 A_␮␯⫽具^HF兩␶_␮^{† exp}共⫺T_g兲关H_g^␬,␶_␯兴exp共T_g兲兩HF典

⫽具^HF兩␶_␮^†exp共␬rd兲exp共⫺T⬘兲

⫻关H_s^␬, exp共⫺␬rd兲␶␯exp共␬rd兲兴exp共T⬘^兲兩HF典^{, 共75兲}

F_␮␯⫽具^HF兩

冉

^1⫹

^兺

^{␭ t¯␭g␶␭†}

冊

^{exp共⫺Tg兲关关Hg␬,␶␮兴,␶␯兴}

⫻exp共T_g兲兩HF典

⫽具^HF兩

冉

^1⫹

^兺

^{␭ t¯␭g␶␭†}

冊

^{exp共␬rd兲exp共⫺T⬘兲}

⫻关关H_s^␬,e^⫺␬rd␶_␮^e␬rd兴,e^⫺␬rd␶_␯^e␬rd兴exp共T⬘兲兩HF典^. 共76兲 Since we here consider states that are invariant with respect to rotations among the occupied orbitals and among the vir- tual orbitals, we may instead of the projection manifold 兵具^HF兩␶_␮^{† exp(}␬rd)其use the manifold兵具^HF兩␶_␮^†其^{. Equation共72兲} and the matrix elements above then reduce to those of the symmetric connection so that the same result is obtained with the general and symmetric connections.

As shown in Ref. 30, the effective Hamiltonian in the symmetric connection H_s is gauge-origin independent.

Gauge independence was the result of the gauge-origin inde- pendence of both the integrals over London orbitals and the connection matrix. It is straightforward to show that, in the symmetric connection, the effective dipole-moment operator is gauge-origin independent as well. As a consequence of the gauge independence of the effective Hamiltonian in the sym- metric connection, the effective ␬s operator is gauge inde- pendent. Hence, H_s^␬ and d_s^␬ are gauge-origin independent, ensuring that the final result is gauge-origin independent.

F. The canonical formulation

We here consider the case where the Hartree–Fock or- bitals are required to be canonical. The␬pq

g matrix now con- tains both redundant and nonredundant orbital rotation pa- rameters, which can be determined by solving the effective canonical equations,