Conditions for invariance of molecular magnetic properties in Landau gauge transformations

General constraints for invariance of magnetic properties in a gauge transformation are analyzed. Sum rules relative to the transformation from Coulomb to Landau gauges are examined in particular. Numerical tests for hydrogen ﬂuoride, water, ammonia, and methane molecule have been carried out in large basis set calculations, using random-phase approximation. The conditions for invariance are severe conditions for accuracy of variational molecular wave functions. © 1995 American Institute of Physics.


I. INTRODUCTION
The quality of an approximate variational wave function, describing a given electronic state of an atom or a molecule, can be assessed a priori, by checking the degree to which certain sum rules are satisfied, independently of any comparison between experimental data and corresponding quantities ͑i.e., electronic properties͒ estimated via the same wave function, which might be misleading in a number of cases. These sum rules are very general quantum mechanical relationships, fully obeyed by exact eigenfunctions to a model hamiltonian. 1 They furnish ''internal'' yardsticks of accuracy as by-products of the main calculation.
In particular, the ability of a variational electronic wave function to predict accurate magnetic properties in a molecule is necessarily related to the degree of gauge dependence of these properties. This is a major physical requirement, as gauge invariance is connected to charge and current conservation in the presence of magnetic field via the continuity equation. 2,3 Within the algebraic approximation, practicality of basis sets for determining theoretical magnetic properties can be estimated a priori, by checking their gauge invariance via proper sum rules.
In addition, a relevant theoretical question might be whether, if the approximate variational wave function is a good one, the gauge transformation also leads to a good approximate wave function. This question can be analyzed within the general framework of unitary invariance. 1 If one uses the same set of trial functions ͕⌿͖, invariant to the action of a unitary operator U ͑which transforms a given function of the set ⌿Ј→⌿ U ЈϭU † ⌿Ј͒ to solve the variational problem for both Hamiltonians H and H U ϭU † HU, then the optimum variational energy stays the same, i.e., Ê U ϭÊ . In addition, the ''best'' variational wave function ⌿ will have a series of physically desirable properties, in that it satisfies certain hypervirial theorems, 1 i.e., the aforementioned sum rules.
In three previous papers, 4 -6 the Landau transformation of the vector potential in the Coulomb gauge has been investigated to obtain formulas for magnetic susceptibility and nuclear magnetic shieldings in a molecule in the presence of a static, time-independent, magnetic field. Quite remarkably, within the Landau gauge the diamagnetic contribution to susceptibility is a diagonal tensor, irrespective of coordinate system; besides, the diamagnetic contribution to nuclear shielding is fully described by a maximum of six independent components in the absence of molecular symmetry.
Numerical results [5][6][7] demonstrate that very accurate electronic wave functions are necessary to obtain paramagnetic contributions to magnetic susceptibility of the same quality as those obtainable within the Coulomb gauge for vector potential. On the other hand, nuclear magnetic shielding tensors in the Landau gauge are characterized by the same quality as those evaluated in the Coulomb gauge. In addition to direct comparison of total magnetic properties within Coulomb and Landau gauges, the accuracy of theoretical estimates can be also checked by analyzing sum rules for origin independence of magnetic properties in a change of coordinate system, which can be described as a gauge transformation of the Landau vector potential. 5,6 These topics have been recently reviewed. 8 The present paper is aimed at deriving, and checking via extended numerical tests, more general constraints for invariance of magnetic properties under a gauge transformation for a molecule in a static homogeneous magnetic field. Besides their theoretical interest, these sum rules, as previously emphasized, can be applied to test the characteristics of excellent basis sets for evaluating magnetic properties. In particular, the sum rules for gauge invariance studied in this paper are helpful to sample a basis set in different regions of the molecular domain, as they involve a series of peculiar operators, able to weigh different portions of charge distribution. Accordingly, an analysis of sum rules for gauge invariance might help understand the conditions under which the gauge transformed wave function is also a good candidate for describing properties of a given electronic state in a molecule.
In Sec. II the general case of an arbitrary gauge transformation is analyzed. Section III deals in particular with sum rules for invariance under a Landau transformation, and corresponding numerical results are discussed in Sec. IV. They lead to insights as to when the Landau gauge transformation will affect the accuracy of the approximations retained in a calculation of magnetic properties.

II. SUM RULES FOR INVARIANCE IN A GENERAL CHANGE OF GAUGE
Let us consider a molecule with n electrons, with mass m e , charge Ϫe, coordinates r i , canonical momenta p i , angular momenta l i ϭr i ϫp i , (iϭ1,2,...,n), and N nuclei, with corresponding quantities M I , Z I e, R I , etc.
The ''particle'' Hamiltonian of the electrons is with eigenstates ͉ j͘ and energy eigenvalues E j (0) ; the reference state is denoted by ͉a͘ ͕the notation of previous papers 4 -6 is retained here, e.g., In the presence of a magnetic field B with vector potential in the Coulomb gauge, the ''interaction'' Hamiltonian is In a gauge transformation of the Coulomb vector potential, induced by the generating function , the unitary operator acting on the electronic wave function is U ϭexp͓Ϫ(ıe/បc)͚ i ͑r i ͔͒. Both for exact eigenfunctions, and in the case of variational eigenfunctions belonging to a set of trial functions invariant to U, 1 the second-order energies, are left invariant. This implies that a calculation of magnetic properties should fulfill constraints which can be expressed in the form of quantum mechanical sum rules. Denoting by total vector potential at r, the contributions to second-order interaction energy can be written

͑11͒
In a gauge transformation ͑6͒ the diamagnetic contributions transform

͑17͒
The paramagnetic contributions transform

͑23͒
Therefore, under a gauge transformation ͑6͒ of the Coulomb vector potential, general conditions for invariance of magnetic susceptibility are obtained via the identities in the form The same formulas are established using the hypervirial theorem, 1 via the off-diagonal relation and the operator equations These results can now be used to work out explicit conditions for invariance of molecular magnetic properties. The magnetic susceptibility tensor within the Coulomb gauge contains paramagnetic and diamagnetic contributions, compare for Eq. ͑7͒, Analogously the paramagnetic and diamagnetic contributions to magnetic shielding of nucleus I carrying the intrinsic moment, compare for Eq. ͑8͒, are

III. SUM RULES FOR INVARIANCE IN A LANDAU TRANSFORMATION
The Landau transformation 9 is induced by the function L ϭ 1 2 ͑ B x yzϩB y zxϩB z xy ͒, ͑40͒ and the Landau vector potential, compare for Eq. ͑6͒, has components A x L ϭA z L ϭ0, A y L ϭB z x, for a magnetic field in The diamagnetic contribution to nuclear shielding is etc. For any coordinate system, xy dIL ϭ yz dIL ϭ zx dIL ϭ0. The paramagnetic contribution is etc. The conditions for invariance imply, according to Eqs. ͑25͒-͑30͒, that the sum rules for susceptibilities, and for nuclear shielding, must be fulfilled ͑other tensor components are obtained by cyclic permutation of the indices x, y, and z͒. In these equations the off-diagonal hypervirial relation for the virial operator, compare for Refs. 1 and 10, has been introduced ͑other symbols have the same meaning as in Refs. 4 -6͒, so that, for instance etc. Sum rules for other components are obtained by cyclic permutations of the indices.

IV. RESULTS AND DISCUSSION
In previous papers 5, 6 magnetic properties have been evaluated ab initio for HF, H 2 O, NH 3 , and CH 4 molecules, within the random-phase approximation ͑RPA͒, using Gaussian basis sets of increasing extension and flexibility.
The quality of theoretical magnetic susceptibilities and nuclear shieldings in the Landau gauge was established by direct comparison with corresponding quantities in the Coulomb gauge. 5,6 Moreover, quite general yardsticks of accuracy for first-order perturbed wave functions, relying on Thomas-Reiche-Kuhn sum rules and other constraints for origin independence of theoretical magnetic properties, were used in extended numerical tests. 5,6 Much more specific criteria for assessing the overall quality of theoretical values in the Landau gauge is provided by Eqs. ͑47͒-͑50͒. According to these constraints, a given basis set should, at the same time, yield accurate representations of V ␣␤ , the virial operator ͑52͒, and of M I␣ n , the operator for the magnetic field of electrons on nucleus I. The former, defined via position and linear momentum, weighs the electron cloud in the tail regions of molecular domain, the latter, owing to ͉rϪR I ͉ Ϫ3 factor, samples charge distribution in the environment of the nuclei. It is quite difficult to meet both these requirements with a Gaussian basis set; ''steep,'' as well as diffuse polarization functions should be necessarily included. Therefore, results reported in Tables I-XV provide additional fairly complete information on the ability of electronic wave functions adopted in previous studies to predict magnetic properties within the Landau gauge. 5,6 For each molecule, four tables, showing theoretical estimates of sum rules ͑47͒, ͑49͒, and ͑50͒ respectively for hydrogen and heavy atom shieldings, are reported in the present study ͓for all the molecules of the series examined here constraint ͑48͒ is satisfied by symmetry͔. For HF, NH 3 , and CH 4 , basis sets I-IV are the same as in Ref. 6 Numerical results relative to sum rule ͑47͒ for susceptibility evaluated assuming the origin on a hydrogen nucleus, compare for Tables I, V, IX, and XIII demonstrate that basis sets of high quality are necessary to guarantee gauge invariance in a Landau transformation. Less accurate estimates were obtained for HF, where the discrepancies between left and right-hand sides of Eq. ͑47͒ are Ϸ10%. On the other hand, the same constraint ͑47͒ is very accurately fulfilled for basis sets IV of NH 3 and CH 4 , and almost exactly satisfied for basis set VI of H 2 O. This trend is confirmed by numerical tests for sum rule ͑49͒ ͑relative to origin on hydrogen nucleus͒ involving two virial operators V ␣␤ compare for Tables II, VI, X and XIV. Possibly, an even more severe probe for wave function accuracy is furnished by this constraint for magnetic susceptibilities, particularly in the case of HF, see Table II. This may imply that the basis sets adopted in the present work are better suited to represent angular momentum operator L ␣ than virial tensor operator V ␣␤ . Accordingly, the Hartree-Fock electronic wave functions adopted in the present study could still be improved to insure a higher degree of invariance in a transformation to the Landau gauge.
Constraint ͑50͒ for magnetic shieldings was checked assuming the origin on the nucleus in question, see Tables III,  VII, XI, and XV for hydrogen and Tables IV, VIII, and XII for the heavy atoms. In the case of carbon shieldings, corresponding sum rule becomes the trivial identity 0ϭ0, exactly fulfilled by symmetry for some tensor components, or virtually satisfied ͑e.g., to three significant figures͒ for other components. Accordingly the relative table is not reported.
Theoretical values calculated via Eq. ͑50͒ for hydrogen are quite good, and possibly close to the Hartree-Fock limit, as can be achieved by inspection. These findings confirm the conclusions, reached in Refs. 5 and 6, that a procedure based on Landau gauge is viable and well suited for accurate a priori determinations of proton magnetic shielding.
Quite different judgments are arrived at by considering theoretical results in Tables IV, VIII, and XII for sum rules checking gauge invariance of the heavy atoms. In the case of fluorine, compare for Table IV, magnitude and sign of right and left-hand sides of Eq. ͑50͒ calculated in the present work are different. For oxygen and nitrogen magnetic shielding similar discrepancies can be observed in Tables VIII and XII. A similar drawback is usually encountered in evaluating the (M I␣ n , P ␤ ) Ϫ1 tensor in analyzing origin dependence of nuclear magnetic shieldings of heavy atoms within the Coulomb gauge. 4 -6 Accordingly, a possible explanation for this partial failure may be partially ascribed to lack of steep p functions in the basis sets retained for heavy atoms. In other words, in order to fulfill sum rule ͑50͒, one could add other sets of p Gaussian functions with high exponents to heavy nucleus basis, possibly forming an even tempered set.

V. CONCLUSIONS
A series of sum rules for gauge invariance of electronic second-order energy terms in a Landau gauge transformation has been worked out. According to a well-known connection between gauge invariance and the continuity equation, 2 the degree to which these constraints are fulfilled provides physical information on the reliability of a calculation of magnetic properties, i.e., charge and current conservation in the presence of magnetic field. These sum rules can be rewritten as hypervirial theorems, 1 which hold exactly for optimum variational wave functions obtained via sets of trial functions invariant to the unitary transformation induced by the Landau change of gauge. Within the algebraic approximation, i.e., expanding the trial functions over a basis set, the constraints for gauge invariance are only approximately obeyed, depending on the quality of the set, i.e., its completeness with respect to various operators involved. Accordingly, the results obtained analyzing the sum rules are useful to sample a given basis set, and furnish clear indications to improve it. In addition, they help understand the conditions that are to be fulfilled to guarantee that, starting from an accurate wave function, the gauge transformed wave function is also a good one. Extended Gaussian basis sets have been adopted for HF, H 2 O, NH 3 , and CH 4 molecules. The calculations show that the sum rules examined in this work yield extremely severe tests of accuracy for SCF wave functions; the basis set must contain diffuse polarization functions to satisfy constraints ͑47͒-͑49͒ ͑see text͒, i.e., an accurate representation of the virial tensor operator should be provided to ensure invariant magnetic susceptibilities.
This requirement is comparably easier to fulfill than that necessary for invariance of magnetic shielding, compare for sum rules ͑50͒. In this case, in addition to guaranteeing a reliable representation of the virial operator, the basis set for heavy nuclei must also be enriched with steep polarization functions to accurately represent the operator for magnetic field of electrons on the nucleus in question.