Formal relations connecting different approaches to calculate relativistic effects on molecular magnetic properties

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I. INTRODUCTION
Relativistic effects on nuclear-magnetic-resonance parameters have shown to be of major importance.In particular, the absolute nuclear-magnetic shielding tensors in heavy atom containing compounds are strongly affected by such effects.Within four-component linear-response theory at the random phase approximation ͑RPA͒ level a series of benchmark results were obtained. 1,2The first ones 1 were obtained within the Sternheim approximation. 3 It has been shown formally 4,5 that the diamagnetic component of magnetic properties is only recovered in the nonrelativistic limit when electron-positron rotations are taken into account.Therefore in the four-component linear-response theory the paramagnetic term is identified with the linear-response related to electron-electron rotations, while the diamagnetic one is identified with the contribution associated to electronpositron rotations.Since the early results of Visscher et al., 1 it has been recognized that electron-positron rotations must be taken into account beyond the Sternheim 3 approximation in order to obtain correct results 2 of the nuclear-magnetic shielding tensor.However there are different definitions of para-and diamagnetic contributions at the four-component level yielding the same para-and diamagnetic separation in the nonrelativistic limit.For instance, recently, Visscher 6 analyzed the implementation and numerical results of the nuclear-magnetic shielding within a method of "minimal coupling" of large and small components proposed by Kutzelnigg. 7In this method a unitary transformation of the four-component Dirac Hamiltonian in the presence of a magnetic potential A is carried out.As a result, the magnetic perturbation is expressed as a sum of a linear and a quadratic term in A, which closely resemble the nonrelativistic operators ͑plus additional terms͒.Rather unexpectedly, however, numerical results of the usual linear-response approach and minimal-coupling approach show significant differences. 6ven though differences in the paramagnetic and diamagnetic terms were to be expected, the total nuclear-magnetic shieldings should be in close agreement.][10][11][12][13][14][15][16] Among them, in the linear response within the elimination of small component ͑LR-ESC͒ approach, 9 the separation into para-and diamagnetic terms is explicitly obtained by considering separately the contributions from electronic excited states on one hand, and contributions from electron-positron pair-creation terms on the other.This decomposition is therefore fully consistent with the one carried out in the four-component linear-response calculations 1,2 which considers separately the contributions from electronic and positronic states to a given positive-energy fourcomponent spinor in order to define the paramagnetic and diamagnetic components.The Breit-Pauli approach, in which the magnetic vector potential A is added to the canonical momentum in the Breit-Pauli Hamiltonian, was first investigated by Fukui et al., 17,18 and later expanded and completed by Manninen et al. 8 and Fukui et al. 18 However, it is interesting to remark the following point.It seems natural to identify the paramagnetic contribution as that originating in operators linear in A, and to identify the diamagnetic one with operators quadratic in A. This definition leads to a different separation into para-and diamagnetic components in the Breit-Pauli approach to those of the LR-ESC one.In fact, this definition applied to the Breit-Pauli approach does not correspond to a decomposition into electron excitations and electron-positron contributions at the four-component level.A formal relation connecting the Breit-Pauli and LR-ESC approaches was proven in Ref. 9. Kutzelnigg 16 developed two-component expressions of magnetic properties starting from the Dirac equation in the presence of a magnetic field and using direct perturbation theory.By carrying out a double perturbation expansion ͑both in c −1 and the magnetic potential͒ he arrived at formal perturbative expressions based on the Lévy-Leblond Hamiltonian.In this way, some singularities of the Foldy-Wouthuysen transformation are avoided.
In the present work, different formal relations allowing to analyze different questions related to the leading-order ESC approach are discussed.The ESC reduction of Visscher's nuclear-magnetic shielding based on Kutzelnigg's minimal-coupling approach is carried out.In this way a comparison of the LR-ESC, Breit-Pauli, and ESC-reduced minimal-coupling approaches can be established.Formal relations connecting the different para-and diamagnetic contributions of each approach are explicitly proven.Numerical results based on calculations of Ref. 10 are presented in order to verify explicitly the fulfillment of such relations.HX and CH 3 X ͑X =Br,I͒ are taken as model compounds.In a second section, the gauge origin invariance of the LR-ESC approach is analyzed.Formal relations connecting different contributions to the nuclear-magnetic shielding for different gauge origins are explicitly proven for all one-body operators of the LR-ESC formalism.Numerical results are presented for the calculation of the H nuclear-magnetic shielding in HX ͑X =Br,I͒, by considering a shift of the gauge origin from the H nucleus to the X nucleus.Results thus obtained fully explain differences found in previous works between the closely related operators of the LR-ESC ͑gauge origin at the H nucleus͒ 10 and Breit-Pauli ͑gauge origin at the heavy nucleus͒ 8,19 approaches.

Breit-Pauli approach
The Pauli Hamiltonian can be obtained as the leadingorder ESC approximation of the Dirac Hamiltonian, i.e., where L and S stand for the large and small components of a Dirac four-component spinor of positive energy mc 2 + E, stand for the 2 ϫ 2 Pauli matrices, p is the momentum operator, and V is the Coulomb potential of the nuclei.In the ESC scheme, and the leading-order relativistic approximation is Inserting this relation in Eq. ͑1͒ the following equation for the large component is obtained that Consistently up to order c −2 , this equation is transformed to an eigenvalue equation for the hermitian Pauli Hamiltonian by means of the following transformation:

͑6͒
The one-body Pauli Hamiltonian takes the form,

͑7͒
where the curly brackets stand for the anticommutator.The first two terms yield the Schrödinger Hamiltonian H. Therefore, relativistic effects on molecular states can be obtained from Rayleigh-Schrödinger perturbation theory ͑RSPT͒ for the Hamiltonian, where H R gathers the well-known relativistic spin-orbit ͑SO͒, Darwin ͑Dw͒, and mass-velocity ͑MV͒ operators.Two-body interaction terms in this approximation are obtained by considering the Coulomb and Breit interaction operators between positive-energy four-component spinors and carrying out the ESC procedure in order to express the leading-order contributions ͑up to order c −2 ͒ in terms of Pauli two-component spinors ˜.This procedure yields the Breit-Pauli Hamiltonian ͑see, e.g., Ref. 20͒.In the present work we restrict ourselves to the analysis of one-body operators and therefore two-body operators are not presented explicitly.
In the presence of a magnetic field described by a vector potential A the operator p in the Dirac Hamiltonian is replaced by operator = p + ͑1/c͒A ͑in a.u.͒.All steps of the previous derivation remain valid and the Pauli Hamiltonian in the presence of a magnetic field is obtained making the same replacement in Eq. ͑7͒.It is obtained that where are the "nonrelativistic" magnetic operators, and yield the leading-order relativistic field-dependent corrections.It is worthy to note that the commutator in Eq. ͑12͒ yields i.e., it is the "field-induced" spin-orbit operator considered in different works. 8,17,21,22As a consequence, relativistic effects on magnetic properties within the Breit-Pauli approximation can be expressed as first-, second-, and third-order RSPT corrections to the Schrödinger molecular energy.

Linear response "LR… within the ESC approach
Within the LR-ESC approach 9 the starting point is an RSPT ͑2͒-like expression of the magnetic properties bilinear in the magnetic potential, based on the Breit Hamiltonian in the four-component Dirac-Fock space. 20,23The perturbation operator is W = ␣ • A ͑where ␣ are the 4 ϫ 4 Dirac vector ma-trices͒.Within the no-pair approximation, the ground state ͉0 N ͘ of an N-electron system can be expressed as a combination of determinants containing only "electronic" fourcomponent spinors, i.e., positive-energy eigenvectors of the one-body Dirac Hamiltonian.The magnetic perturbation can connect such state with excited states of the same type, ͕͉n N ͖͘, and with states which have an electron-positron pair created on ͉0 N ͘.Considering contributions from different particle number manifolds in Dirac-Fock space, the para ͑E para ͒ and diamagnetic ͑E dia ͒ contributions can be defined as 9 The last term in E dia takes account of a renormalization of the vacuum energy due to the magnetic potential.This separation into para-and diamagnetic contributions is consistent with that of Refs. 1, 2, 4, and 5 in the context of the linearresponse four-component ͑RPA͒ approach.In the LR-ESC approach the lowest-order relativistic corrections are obtained by applying the leading-order ESC approximation to all terms in Eqs.͑15͒ and ͑16͒.As a result a set of operators describing relativistic effects on magnetic properties is obtained. 9,10Magnetic-field-dependent operators entering the "paramagnetic" contribution, Eq. ͑15͒, are obtained when the matrix elements of the magnetic interaction between positive-energy four-component spinors are considered where, consistently in the leading-order ESC scheme it holds

͑18͒
and the relation between the large component and the twocomponent spinors of the unperturbed Pauli Hamiltonian is

͑19͒
Explicit third-order field-dependent operators were obtained in Ref. 9 from the relation Eq. ͑17͒.A brief account is given here.The following relation is correct up to order c −3 : Taking Eqs.͑17͒-͑20͒ into account, the following operators can be defined as where the superscript LR indicates that this is the form of the field-dependent operator in Ref. 9.
In addition to the nonrelativistic diamagnetic operator, the leading-order relativistic correction to the "diamagnetic" term was obtained by expressing the electron-positron contribution to magnetic properties, Eq. ͑16͒, in terms of Pauli two-component spinors consistently with the ESC approach.The detailed derivation of the resulting operators is presented in Ref. 9. A brief summary is presented in Appendix A. As a consequence, the following operator bilinear in the magnetic potential was obtained: where the superscript LR indicates that this is the form of the diamagnetic operator in the LR-ESC scheme.

Leading-order ESC-reduced "minimal-coupling" approach
Recently, a unitary transformation of the Dirac Hamiltonian in the presence of a magnetic potential yielding a minimal coupling between the large and small components of four-component spinors was proposed by Kutzelnigg. 7he unitary operator: is applied to the Dirac one-body Hamiltonian in the presence of a magnetic potential A, in order to obtain the minimal-coupling-transformed Hamiltonian, It was shown in Ref. 7 that expansion of H K in powers of the magnetic potential up to second-order yields the following Hamiltonian: where H 0 D is the unperturbed Dirac Hamiltonian ͓A = 0 in Eq. ͑25͔͒, and In this way, the odd terms ␣A are removed from the Hamiltonian.Within this approach, magnetic properties bilinear in the magnetic potential are obtained from first-and secondorder four-component perturbation theory considering operators D ͑2͒ and D ͑1͒ , respectively, i.e., where ͗͗A ; B͘͘ is a shorthand notation for a second-order correction to the molecular energy and the second term is an expectation value for the unperturbed molecular relativistic ground state.This approach was implemented by Visscher for the calculation of the magnetic shielding tensor 6 within the four-component linear-response RPA formalism.The contributions to the molecular energy given in Eq. ͑30͒ are related to the second-order expression for the usual magnetic interaction, 1 by a formal relation provided by the proposed unitary transformation itself.At the four-component level, this relation is as follows:

͑33͒
If the leading-order ESC approximation ͓Eqs.͑17͒-͑20͔͒ is applied to operators D ͑1͒ and D ͑2͒ , the following linear and quadratic operators are found where the superscript K indicates that these are the ESCreduced operators of the minimal-coupling formalism.A brief derivation of these operators is presented in Appendix B, as a detailed account of leading-order effects can be found in Ref. 7.

B. Relations connecting the Breit-Pauli, LR-ESC, and ESC-reduced minimal-coupling approaches
Within the leading-order ESC schemes presented in the previous sections, relativistic effects on magnetic properties which are bilinear in the magnetic potential can be expressed as first-, second-, and third-order RSPT corrections to the Schrödinger molecular energy.Two kinds of relativistic effects can be distinguished: ͑i͒ "passive" effects, in which relativistic corrections to the unperturbed molecular states are considered by means of the spin-orbit, Darwin, and massvelocity operators in H R , Eq. ͑8͒, and lowest-order magnetic operators O 1 , and O Diam,NR ͓Eqs.͑10͒ and ͑11͔͒; and ͑ii͒ "active" effects, in which the explicit field-dependent operators O 3,X and O Diam,X ͑X = BP, LR, or K͒ are considered.This terminology has become conventional in the bibliography, 8,19 and is therefore applied in the present work.
Passive effects are exactly the same in the Breit-Pauli, LR-ESC, and minimal-coupling formalisms.Active effects are obtained by considering, on one hand, second-order RSPT corrections to the molecular energy for operators O 1 and O 3,X , where O͑N͒ stands for the one-body operator A in the N particles states space On the other hand, the relativistic active correction to the diamagnetic contribution within each formalism is obtained as a first-order RSPT energy correction for the corresponding O Diam,X operator, i.e., In order to analyze the relations between the different ESC approaches for the calculation of molecular magnetic properties, the commutator in Eq. ͑39͒ is considered where H is the many-body Schrödinger Hamiltonian.The second-order RSPT correction to the molecular energy involving a commutator with H can be reexpressed as i.e., as an expectation value for an operator bilinear in the magnetic potential.Results Eqs. ͑39͒ and ͑40͒ show that by means of these formal relations, paramagnetic contributions to magnetic properties involving operator O 3 ͑H͒ can be transformed into diamagnetic contributions, i.e., contributions which are obtained as an expectation value for an operator bilinear in the magnetic potential.Explicit relations among the different ESC formalisms are hereby proven.
By a simple rearrangement of terms in Eq. ͑22͒ O 3,LR can be expressed as and, therefore, ͓see Eq. ͑12͔͒ where U stands for the two-body Coulomb interaction in the Schrödinger Hamiltonian.The secondorder RSPT expression involving the third term in Eq. ͑42͒ can be taken into account as indicated in Eq. ͑40͒, yielding a modified diamagnetic operator of the form, i.e., the operator of the Breit-Pauli scheme, Eq. ͑13͒, is readily obtained.Therefore, we have obtained a formal expression relating the operators of the LR-ESC scheme and those of the Breit-Pauli one. 9In Eq. ͑42͒ there is a two-body contribution which should be taken into account in order to make the relation between both approaches exact.Such contribution is neglected in the present work.It is seen that the separation into paramagnetic and diamagnetic contributions associating the linear operators to the first one and the quadratic to the second in the Breit-Pauli approach does not correspond to the separation into electronic excitations and electron-positron rotations, as does the LR-ESC scheme.In the present work the fulfillment of these relations is analyzed numerically taking HBr and HI as model examples.
The ESC-reduced minimal-coupling operators can also be obtained starting from the LR-ESC ones, Eqs.͑22͒ and ͑23͒, and reexpressing them in a convenient way.The term containing V in O 3,LR in Eq. ͑22͒ can be replaced in terms of O 3 ͑H͒, Eq. ͑39͒, yielding the following formal expression: Taking into account that the second and third terms in Eq. ͑44͒ can be written as where O 3,K is the ESC-reduced operator of Eq. ͑34͒.Once again, the contribution of the second term of Eq. ͑46͒ to the molecular energy can be reexpressed as an expectation value for the operator in Eq. ͑40͒.Therefore the form of the quadratic operator in this approach is i.e., precisely the ESC-reduced operator of Kutzelnigg's formalism, Eq. ͑35͒.It is interesting to remark that in this context there are no operators involving the one-body Coulomb potential V.

C. The nuclear-magnetic shielding tensor in the Breit-Pauli, LR-ESC, and ESC-reduced minimal-coupling approaches
The NMR nuclear-magnetic shielding tensor can be calculated as , where E is the molecular energy in the presence of the magnetic moment of nucleus N, N , and a uniform magnetic field B. The lowest-order relativistic corrections to the magnetic shielding tensor of nucleus N arising from passive operators are the same for all approaches, as they involve the nonrelativistic magnetic operators and relativistic effects are only included in the molecular magnetically unperturbed states.Relativistic corrections due to active operators within the different approaches considered are obtained from the RSPT expressions bilinear in N and B, i.e., where the last term represents an expectation value for the Schrödinger molecular ground state, X stands for the Breit-Pauli ͑X =BP͒, LR-ESC ͑X =LR͒, or minimal-coupling ͑X = K͒ approach, and gathers the paramagnetic ͑nuclear͒ spin-orbit ͑PSO͒ and Fermi contact ͑FC͒ and spin-dipolar ͑SD͒ operators, where r N is the electron position with respect to nucleus N; and contains the orbital Zeeman ͑OZ͒ and spin Zeeman ͑SZ͒ operators, In this section, LR-ESC operators will be considered in first place.The only third-order operators contributing to the nuclear-magnetic shielding tensor are

͑55͒
hereafter identified as PSO-K ͑it combines the PSO and kinetic-energy operators͒ identified as OZ-K, and where the first two terms are dubbed SZ-K and the last one is dubbed BSO ͑field-induced spin-orbit͒.The superscript S and T in Eqs.͑55͒-͑57͒ stand for singlet and triplet operators, respectively.As indicated in Sec.II B the relation between different approaches is based in the subtraction of a term proportional to O 3 ͑h͒ to the LR-ESC operators ͑where h stands for the one-body part of the Schrödinger Hamil-tonian͒.O 3 ͑h͒ is a purely triplet operator, and therefore the different approaches yield different expressions for operator O 3T ͑B͒, while all other operators Eqs.͑50͒, ͑53͒, ͑55͒, and ͑56͒ remain unchanged.Explicitly, we have where O 3 ͑h , B͒ is the commutator of Eq. ͑39͒ specialized to the case of the uniform magnetic field.Consistently with results in Eq. ͑40͒, it holds where O 3 ͑U , B͒ is the commutator Eq. ͑39͒ with the twobody Coulomb potential U and the uniform magnetic-field vector potential, and On the other hand, O Diam ͑ N ; B͒ collects terms bilinear in N and B from the quadratic operator O Diam .Within the LR-ESC approach it is obtained that For the sake of clarity, each term in Eq. ͑61͒ will be referred to as W i LR ͑i =1,4͒, respectively, in order to be able to identify each contribution separately.For a real ground state ͑in coordinate representation͒ it holds where A B ͑R N ͒ is the vector potential of the uniform magnetic field at the position of nucleus N and vanishes if the gauge origin is placed at the nucleus site.The expectation value in Eq. ͑59͒ contains exactly the operator Eq. ͑62͒.
In Table I we present a summary of the contributions of active operators to both the paramagnetic and diamagnetic terms of the nuclear-magnetic shielding tensor in the LR-ESC, Breit-Pauli, and ESC-reduced minimal-coupling approaches.The numerical factors indicate the scaling of a given term with respect to its value in the LR-ESC method.Second-order contributions are identified by the two operators combined in E͑A , B͒ and first-order ones are separated into two terms: 1 ͑Dia− K͒ collects those terms containing a delta-type operator at nucleus N ͓W 2 and W 3 in Eq. ͑61͔͒; and 2 ͑Dia− K͒ contains contributions from W 1 and W 4 of Eq. ͑61͒.
In Table II II in a simple way ͑not even the dia-and paramagnetic separation refer to the same terms͒.The minimum coupling of large and small components in Kutzelnigg's approach yields the smaller ͑in absolute value͒ paramagnetic and diamagnetic relativistic corrections.The problem of the inconsistency between the fourcomponent calculations based on the standard RPA formalism 1,2 and Kutzelnigg's minimal-coupling one is not present at this level of approximation and confirms the suspicion that the difficulties are related to the positron rotation terms as considered in the four-component RPA calculations.
As it can be seen in Tables I and II the equivalence between the different methods relies on the following identity, extracted from Eq. ͑40͒ It is seen that in the right-hand side ͑rhs͒ of Eq. ͑63͒ there is a two-body contribution, which is automatically taken into account in the left-hand side ͑lhs͒ in terms of one-body operators only.Such two-body contribution must be present in the context of the Breit-Pauli approach.It represents the two-body part of the second-order spin-orbit operator found in Refs.17 and 22.Unfortunately, no numerical estimates of this two-body term are found in the bibliography for the systems under study in the present work.It is interesting to emphasize that such contribution is automatically contained in the LR-ESC approach as part of calculations involving only one-body operators.As it is seen in Eq. ͑63͒, such contribution is needed to make the identity exact.Therefore, the difference in the total relativistic correction calculated with the LR-ESC, Breit-Pauli, and ESC-minimalcoupling approaches retaining only one-body operators has  two sources: on one hand, the lack of the two-body term just mentioned and, on the other hand, differences due to the incompleteness of the basis set.The difference of 28.71 ppm found for I in HI between different approaches arises from these sources of error, and it can be considered very small compared to the total relativistic effect of ca.1000 ppm.
In Table III the verification of the identity Eq. ͑63͒ is explicitly checked for X in HX and X and C in CH 3 X ͑X =Br,I͒.Values presented correspond to the right-hand side of Eq. ͑63͒ compared to those of the left-hand side of Eq. ͑63͒, neglecting the two-body contribution.Numerical values are taken from Ref. 10.
The fact that the difference is larger for the heaviest nucleus, I, significant for Br, and negligibly small for C could be an indication that it is really a measure of the importance of the two-body term and basis set incompleteness errors are negligibly small.

D. Gauge origin invariance of the LR-ESC approach
When considering magnetic molecular properties, the problem of the gauge origin of the vector potential of the uniform magnetic field must be addressed, as quantities of physical significance must be gauge origin invariant.In non relativistic theory, the numerical values of the paramagnetic and diamagnetic terms of molecular properties depend on the gauge origin considered, but the total value of the magnetic property under study is gauge origin independent.The cancellation of the change in the paramagnetic and diamagnetic terms is demonstrated formally proving the existence of a formal relation connecting the contribution to the energy of operators containing the shift from one gauge origin to the other.
In the present section the gauge origin invariance of the LR-ESC nuclear-magnetic shielding is analyzed.All onebody operators yielding both active and passive corrections are considered.The superindex LR is dropped from all operators for brevity.Two-body terms are not taken into account, as no actual calculations have been carried out for them.Active relativistic corrections are summarized in Eq. ͑49͒.Passive relativistic effects are obtained by considering relativistic corrections to the unperturbed molecular states in the presence of the nonrelativistic magnetic operators, i.e., where E͑A ; B ; C͒ represents a third-order RSPT energy correction, and H R represents the relativistic corrections to the Schrödinger Hamiltonian as given in the Pauli Hamiltonian, The first term is the mass-velocity operator ͑MV͒, the second one is the Darwin operator ͑Dw͒, and the third one is the spin-orbit operator ͑SO͒.In Eq. ͑66͒, only the one-body terms of H R are retained, in order to be consistent with the purpose of obtaining connections between one-body operators only.
If a change of gauge origin of the uniform magnetic potential is carried out, given by a displacement vector R G , the change in the individual contributions of Eqs.͑49͒ and ͑64͒ can be expressed in terms of the change in A B , where h is the one-body Schrödinger Hamiltonian, The difference in the contribution of each term can be expressed as
In Eqs.͑71͒ and ͑73͒ the commutator of Eq. ͑68͒ is involved.In order to make use of Eqs.͑75͒ and ͑76͒, the one-body Hamiltonian h should be replaced by the manybody Hamiltonian H.This can be done without the appearence of extra two-body terms, because the operator r in Eq. ͑68͒ commutes with the two-body Coulomb interaction operator.Therefore, making use of Eqs.͑75͒ and ͑76͒ the following relations are found to hold exactly.
In the case of Eq. ͑71͒ it is obtained that ͑77͒ Details of the derivation are presented in Appendix C ͑identity 1͒.In the case of Eq. ͑73͒, it is useful to separate it into two terms: one, H R,S , containing the scalar relativistic massvelocity and Darwin operators, and a second one, H R,T , containing the spin-orbit one ͑SO͒ ͑see Appendix C, identities 3 and 4͒, In the rhs of Eq. ͑78͒ the change in the contribution from singlet operators of Eq. ͑69͒, Eq. ͑70͒, is already included.Therefore, only the term containing triplet operators in Eq. ͑70͒ must be considered explicitly.To this end it is useful to observe that the triplet operator in Eq. ͑69͒ can be expressed as the following commutator:

͑80͒
In order to apply Eq. ͑75͒, the one-body Hamiltonian h must be replaced by the many-body Hamiltonian H.The difference between them is given by the two-body Coulomb interaction and leads to a two-body contribution to Eq. ͑70͒.Such term is neglected in the present work.Therefore, application of Eq. ͑75͒ yields ͑see Appendix C, identity 2͒

͑81͒
In Table IV the changes in different contributions and the cancellation of all terms showing the gauge origin invariance of the full one-body relativistic correction to the nuclearmagnetic shielding are summarized.
In numerical results of relativistic corrections to the nuclear-magnetic shielding within different approaches, different gauge origins were used for the calculation of the nuclear-magnetic shielding of the H nucleus in HX compounds. 8,10,19It is thus interesting to compare numerically the fulfillment of the formal relations found in the present work connecting results with different gauge origins.We present numerical values taken from Ref. 10.As the formal fulfillment of such relations was proven, two sources of error can be expected.On one hand, numerical errors due to incompleteness of the basis sets used in the calculations, or numerical errors introduced due to the approximate procedure used to calculate matrix elements of some operators in Ref. 10. On the other hand, differences may also be expected whenever the fulfillment of a given formal relation depends on contributions from two-body operators.This is the case of Eq. ͑81͒.Numerical results are presented in Table V for HBr and in Table VI for HI.
From Tables V and VI the following conclusions can be obtained.All relations are fulfilled to a large extent in both cases.When small differences are found, they are surely due to basis set incompleteness errors, as they are of the same order of magnitude as the differences found in final results obtained with different basis sets in Ref. 10.The relation in Eq. ͑81͒ holds almost exactly.This feature indicates that two-body effects are really small in these calculations.When the gauge origin is changed from the X center to the H one in the calculation of the H nuclear magnetic shielding, the largest changes are found for the scalar third order, the OZ-K, the PSO-K, and "active" diamagnetic terms W i .All these terms yield negligibly small results when the gauge origin is placed at the X nucleus. 8,19When the gauge origin is placed at the H nucleus, all these terms take non-negligible values that cancel each other, confirming the importance of taking them all into account in a consistent way.

III. CONCLUDING REMARKS
In this work a series of formal relations connecting different approaches to calculate relativistic effects at the leading-order ESC level of approximation were proven.In addition, the gauge origin invariance of the LR-ESC theory has been established.It is worthy to mention that although the main goal has been to establish these relations theoretically, numerical results are also presented in order to verify their fulfillment explicitly in practical applications.As a consequence several aspects of the relativistic calculation of nuclear-magnetic shieldings at the ESC level have been clarified.In particular, differences in numerical results of the LR-ESC and Breit-Pauli approaches in HI are explained for both the heavy and H nuclei. Full consistency between the LR-ESC, Breit-Pauli, and ESC-minimal-coupling approaches was found.Differences in numerical values found at the four-component level are not present within the ESC approximation.A deeper comparison between the LR-ESC and minimal-coupling methods would be possible by carrying out explicitly the ESC reduction of each individual contribution of the nuclear-magnetic shielding in Ref. 6.A comparison with Kutzelnigg's direct perturbation-theory approach 16 would also be very interesting, as final expressions of such formalism are very different to ESC ones.Work along these lines is being carried out by our research group.

ACKNOWLEDGMENTS
Financial support from UBACYT ͑Project No. X222͒ and CONICET ͑Project Nos.PEI 6217 and PIP 5119͒ is gratefully acknowledged.Two of the authors ͑M.C.R.J.A. and C.G.G.͒ are members of "Carrera del Investigador," CONICET.One of the authors ͑J.I.M.͒ has a fellowship grant from Universidad de Buenos Aires.

APPENDIX A: THE DIAMAGNETIC TERM IN THE LR-ESC APPROACH
The starting point is the pair-creation contribution E dia , Eq. ͑16͒.Consistently up to order c −4 , the following approximation holds with similar expressions for the term involving the vacuum state in Eq. ͑16͒.In Eq. ͑A1͒ the first ͑second͒ term is of order c −2 ͑c −4 ͒ or lower.
These expectation values can be evaluated as follows: ͑␣AP p ␣A͒ e,e Ј e + eЈ͉0 N ͘, ͑A4͒ where the second-quantized forms of the operators was used, e͑p͒ stand for "electronic" ͑positronic͒ four-component Dirac spinors, and O ep = ͗e͉O͉p͘ represent matrix elements of one-body operator O. Similar steps can be carried out for the second and fourth terms in Eq. ͑A2͒.It is concluded that E dia can be expressed as an expectation value for the following one-body operators: where the first term yields the diamagnetic term in the nonrelativistic limit.In order to carry out the leading-order ESC reduction of Eq. ͑A5͒, the projector P p in the space of fourcomponent Dirac spinors ͑splitted into 2 ϫ 2 blocks͒ can be written as where x = p /2mc.Therefore where only terms up to quadratic in x were retained.In order to evaluate the expectation values in Eq. ͑A5͒ it is necessary to consider expectation values of all operators between positive-energy four-component spinors, where only terms up to order x 2 need be retained, and the expectation values are now to be carried out for the twocomponent Breit-Pauli spinors.Summing up, Eq. ͑A5͒ is reexpressed as where O Diam,LR = − 1 8m 3 c 4 ͑͑pA͒ 2 + 2Ap 2 A + ͑Ap͒ 2 + ͕p 2 ,A 2 ͖͒, ͑A11͒ as indicated in Eq. ͑23͒.Due to the factor c −4 , the corresponding expectation value must be carried out for the nonrelativistic ͑i.e., Schrödinger͒ ground state.

APPENDIX B: ESC REDUCTION OF OPERATORS OF THE MINIMAL-COUPLING APPROACH
The ESC reduction of operators of the minimal-coupling approach is as follows.In the case of the linear operator Eq. ͑28͒, matrix elements between different four-component spinors are considered Consistently to order c −3 , making use of Eqs.͑17͒-͑19͒, Eq. ͑B1͒ can be expressed as yielding the final result, Eq. ͑34͒.
In the case of the bilinear operator Eq. ͑29͒, the expectation value is considered.

͑B5͒
Inserting the relation of the small and large components with the Pauli two-spinors, Eqs.͑17͒-͑19͒ consistently to the order c −3 the operator Eq. ͑35͒ is obtained.

APPENDIX C: FORMAL RELATIONS PROVING THE GAUGE ORIGIN INVARIANCE OF THE LR-ESC APPROACH
Formal relations for gauge invariance are hereby proven.
Diam,X .Values obtained by the LR-ESC ͑X =LR͒, Breit-Pauli ͑X =BP͒, and ESC minimal-coupling ͑X =K͒ approaches ͑Numerical values taken from Ref. 10 and rescaled according to numerical factors in Table I͒.Gauge origin at I. Values in ppm.
cOperators W 1 X are defined after Eq. ͑61͒.TABLE II.Relativistic corrections to the nuclear-magnetic shielding constant ͑I͒ in HI.E͑A , B X ͒: RSPT͑2͒ expressions involving operators A and B X ; W i X ͑i =1,4͒: expectation values for operators O a See footnote to Table I for operators acronyms.

TABLE III .
Numerical verification of relations connecting the LR-ESC, Breit-Pauli, and ESC-minimal-coupling formalisms.As a consequence of such relation the numerical values of the lhs and rhs of Eq. ͑63͒ must be equivalent, except for a two-body contribution.All values in ppm.

TABLE IV .
Relations among the gauge origin dependence of different terms yielding the relativistic correction to the nuclear-magnetic shielding tensor in the LR-ESC approach.The cancellation of all terms in the last column shows the ͑formal͒ gauge origin independence of the ͑one-body͒ total result.
H B stands for the Breit Hamiltonian in the Dirac-Fock space.For the purpose of the present work, H B can be replaced by the one-body Dirac hamiltonian H 0 ͑A2͒Within the approximation Eq. ͑A1͒, the intermediate states in E dia act as a projection operator on the N + 2 particles state space P N+2 ͑and two-particle states for the term involving the vacuum, P 2 ͒.Therefore it is obtained that

TABLE V .
Gauge origin dependence of the LR-ESC relativistic corrections to the nuclear-magnetic shielding of H nucleus in HBr.Numerical values correspond to the difference in a given contribution when the gauge origin is shifted from the Br nucleus to the H nucleus.

TABLE VI .
Gauge origin dependence of the LR-ESC relativistic corrections to the nuclear-magnetic shielding of H nucleus in HI.Numerical values correspond to the difference in a given contribution when the gauge origin is shifted from the I nucleus to the H nucleus.