Self-energy Effects on Nuclear Magnetic Resonance Parameters within Quantum Electrodynamics Perturbation Theory

A theory for the calculation of self-energy corrections to the nuclear magnetic parameters is given in this paper. It is based on the S-matrix formulation of bound-state quantum electrodynamics (QED). Explicit expressions for the various terms of the S-matrix are given. The interpretation of the self-energy, one-and two-vertex terms and some perspective for possible future developments are discussed.


Introduction
Relativistic effects are at the moment starting to be included routinely in calculations of magnetic molecular properties on molecular systems that contain heavy atoms.That is because we are aware of that they are very important in order to predict experimental results.
The usual treatment of molecular properties by theoretical methods is based on classical electrodynamic theory, i.e without resorting to quantized fields.The reason is that most of the relevant terms are included in the perturbed Hamiltonian just using classical fields.Given that the experimental results for magnetic molecular properties have an increasing precision that is in average less than a few percent of its total value it would be important to relax the limitation of using classical fields in our calculations in order to include additional corrections.This is also important from a formal point of view.
Then the next step for precise calculation of molecular properties is related with QED effects.There were some preliminary studies for atomic systems by different groups.Labzowsky, Goidenko and Pyykko found that the bound-state QED contributions to the g-factor of valence ns electrons in alkali metal atoms is as large as 10 % for Rb and less for the other atoms of that series. 1 They also showed that the radiative corrections for the heavy and superheavy atoms can rise up to 0.5% of the ionization energy. 2 They included QED derived potentials on usual perturbation theory to calculate energy corrections.
Our aim in this paper is to go an step forward trying to describe in some detail a formalism necessary to use when one wants to include QED effects on molecular magnetic properties; even though we have recently worked on the NMR-J spectroscopic parameter by using a different methodology. 3The new scheme presented here can be applied to both spectroscopic parameters mentioned above and is more systematic than the previous one.
In order to include QED effects in calculation of atomic or molecular properties it is possible to work with different formalisms.The formulation of QED we choose to work with is the adiabatic S-matrix approach of Gell-Mann, Low 4 and Sucher 5 which was first applied to the bound-state QED by Labzowsky 6 and later on by Mohr. 7 This formalism was applied very recently to atomic and highly ionized systems. 8,9 n our previous work on this field we worked out QED corrections to the NMR-J spectroscopic parameter but without inclusion of radiative corrections which are now treated properly from the outset.
In the next section we give a summary of quantum electrodynamics perturbation theory based on S-matrix formalism.Then we apply that general formalism to the calculation of self-energy corrections to NMR properties.Depending on the external potential we choose (arising from the nuclear magnetic moment or the external static magnetic field) we arrive to J or σ.

Quantum electrodynamics perturbation theory
We briefly outline in this section the theory for the calculation of level shifts of bound state electrons interacting with quantized radiation.A more comprehensive description can be found elsewhere. 7Bound state quantum electrodynamics is a reformulation of standard QED for freeparticles as described in textbooks. 10It is assumed that Dirac equation with a given classical potential can be solved.The factors α are the 4 × 4 Dirac matrices which, in the standard representation, has the form σ are the Pauli matrices, and 0 and 1 represent the null and unit 2 × 2 matrices.It is convenient to use the set of matrices γ 0 = β and γ = βα in order to manifest explicitly the relativistic covariance properties of the Dirac matrices.
The eigenfunctions φ i are taken as the zeroth-order wave functions in terms of which the field operators are built where a i (b † j ) is the annihilation (creation) operator for an electron (a positron) in the state φ i (φ j ) of energy ε i > 0 (ε j < 0).The Fock operators satisfy the usual anti-commutation relations and zero otherwise.The interaction between electrons and a quantized electromagnetic field is accounted for through the interaction Hamiltonian where j µ (x) = −e ψ(x)γ µ ψ(x) is the Dirac 4-current written in terms of the fermion fields and the γ matrices γ µ = (γ 0 , γ), A µ is the electromagnetic 4-vector potential and x = (x 0 , x) is the position 4-vector 1 .
For the level shifts calculations, it is convenient to replace the interaction Hamiltonian by an adiabatically damped one In that case, Gell-Mann, Low 4 and Sucher 5 have shown that the energy shift of an unperturbed state |0 is given by where S ,λ is the S-matrix defined as i.e. product of Fock operators with the creation operators placed to the left of the annihilation operators times photon and fermion propagators (represented by the underbraces) defined as The time dependence of the propagators can be made explicit if we write them as follows The fermion propagator S F (x, y; E) has the following spectral representation 7 where δ is an infinitesimal positive quantity and n runs over the complete spectrum of eigenfunctions.Expanding ∆E 0 in power of λ, the energy formula can be written from first to fourth order in the S-matrix as It has been shown 7 that, for classical external potentials V and one-electron atoms in the state a, the first and second order level shift formula reduces to the well known expressions from standard perturbation theory As an application of the third order formula, Blundell et al. 8 calculated self-energy corrections in atomic systems in the presence of an external potential.In the next section, we use the fourth order energy expression ∆E 0 to obtain self-energy corrections to the NMR parameters.

Self-energy effects on NMR properties
In this section, the S-matrix theory outlined in the previous section is applied to obtain expressions for the self-energy corrections to the NMR parameters J and σ.The self-energy correction corresponds to the level shifts due to the interaction of an electron with itself via one photon exchange (Fig. 1).For the problem we are tackling, we need to consider this type of diagrams in addition to the interaction with two external potentials, namely, those coming from the interaction of an electron with the magnetic moments µ of two different nuclei (in the case of the tensor J) or the electron with a nucleus and the external magnetic field B (in the case of the nuclear magnetic shielding σ).Hence, we consider the electromagnetic 4-potential as the sum of a quantized potential A µ plus a classical one (ϕ, 0, 0, 0) and take the Dirac-Fock ground state |DF as the unperturbed state |0 .Then, the interaction Hamiltonian splits in where and V (x) = eγ 0 ϕ(x) = γ 0 U (x).In order to discuss simultaneously both J and σ we assume where such that the proper potential should be taken for the property of interest.
Since we are interested in properties quadratic in the external potential V , the ∆E 0 expression can be written as where S (1) Note that S (2a) and S (2b) represent the terms of the second order S-matrix containing only quantized and classical potentials, respectively.It should be also stressed that even when we are treating a many-electron system, the use of the approximation of independent particles and, hence, the lack of dynamical correlation in the wave functions, allows us to use all the formal machinery devised for one-electron calculations.
Let us consider separately the different contributions to the S-matrix.

First order
As an illustration of the use of the level shift formulas, let us consider in detail the calculations of the first few orders of the S-matrix.To first order, writing explicitly the time dependence, and taking into account that the state |0 has no positrons, we get For the Dirac-Fock state |0 , : a † i a j : = δ ij , where i must be an occupied state, say, α.Then, that is, ∆E 0 = occ α U αα , which is a generalization of the one-electron case.

Second order
In this section we shall consider in some detail the calculation of the terms S (2a) λ and S (2b) λ , because their treatment involves some general procedures to be applied in the more complicate higher order terms.

Term S (2a) λ
Application of the Wick theorem to the product T {H A I (x)H A I (y)} gives one-and two-electron operators.The one-electron term 2γ µ D F (x, y)S F (x, y)γ µ : ψ(x)ψ(y) : is related to the self-energy of the DF-occupied states and is the only term to be included in S (2a) λ .The two-electron term is not relevant for the calculations involved here since it gives a contribution related to one-photon exchange interaction between two-electrons, namely, the Coulomb-Breit interaction.As in the first order case, the expectation value : ψ(x)ψ(y) : becomes a summation over occupied states α.Integration over time variable x 0 gives and a similar result for the integration over y 0 , where Recalling that at the end of the calculations we take → 0, the roots u = 0 of the argument of ∆ (u) determine the most relevant energy regions for the integrals over E. So, for S (2a) λ , the energy denominator emphasize the region E = ε α − k 0 and we can approximate the integral over E by evaluating the propagator S(x, y; E) at that energy value.Hereafter, this will be a general strategy for the evaluation of those integrals.Hence, where the self-energy insertion Σ mn (ε) is defined by that is,

Term S (2b) λ
The time-ordered product T {H B I (x)H B I (y)} has also contributions from one-and two-electron operators Inserting Eq. (39) into the expression (28) for S (2b) λ , the integration over the time variables gives for the first (one-electron) term; while for the two-electron one.Furthermore, for the DF ground state, a † i a j = δ ij and a † i a † k a l a j = δ ij δ kl − δ il δ kj , j and l being occupied orbitals, say, α and β.Hence, the ∆ −functions emphasize, gives where we have replaced the fermion propagator by its spectral representation, Eq. ( 14).This is a common technique used throughout this paper, and we use it repeatedly in the following sections, for the calculation of higher order terms.It should be noted that the first term of the previous Eq. is of order O( 2 ) because α = β.Therefore, it does not contribute to the energy shift and, hereafter, it will be we disregarded. Finally, where l α (l α ) represents states degenerate (non-degenerate) with α.

Third order
Third order S-matrix, Eq. ( 29) produces one-and two-electron operators.The former comes from the diagrams depicted in Fig. 2, while the later are product of S (1) times S (2a) , i.e., a diagram with two electron lines, one of them interacting with the external field and the other having a self-energy graph (Fig. 1).Diagrams of Fig. 2 have been previously considered 8 in the calculation of self-energy corrections in atomic systems in the presence of an external potential.They were named left-side (L), right-side (R) and vertex (V) terms.Following the same lines as the calculation of first and second order terms, we get expressions which represent many-electron generalizations of those from Ref.
The ∆ -functions favor L and S (3) R , respectively.So, if we proceed, as in the previous lower orders calculations, to replace the fermion propagators by its spectral representations, the whole expression diverges.However, S (3) V does not suffer of this problem because the arguments of the ∆ -functions does not vanish at any energy eigenvalue ε i .Then, a special treatment must be given to the side terms, while S (3) V can be handled as before to give where the vertex insertion Λ nm (ε) is defined by Inserting those terms of the spectral representation of the fermion propagators, Eq. ( 14), which are non degenerate with the state α, the calculations follows the same steps of the vertex term and we get A similar expression for the right term is To handle those terms of Eq. ( 14) corresponding to states degenerates with α, we expand the second propagator in power series around E = ε α − k 0 to first order; for example, in the left side term where The first term of Eq. ( 51), combined with the part of the spectral representation of the propagator (14) containing states α degenerate with α Finally, the last term of the expansion (51), together with those from the representation (14) corresponding to states α , give rise to the so-called derivative terms 7 and It is worth to mention the symmetry between both side terms.

Fourth order
Following the same procedure as before, we get the following contributions to S (4) corresponding to the Feynman diagrams of Fig. 3, where and M i (i = V V, LV, V R, LR, LL, RR) is defined as follows The subscripts (L), (R) and (V) have the same meaning as those from third order terms S (3) .Integration over time variables gives products of ∆ -functions emphasizing the energy regions detailed in Table 1.As it is shown there, the relevant point for the energy landscape in the V V term correspond to the value ε α − k 0 which is not a pole for any of the propagators.Hence, the term S  Table 1: Energy regions emphasized by the ∆ -functions after integration over time variables in the contributions to S (4) .See text after Eq. (58) Term Leading energy regions calculated after some long but straightforward manipulations, to give The other contributions, however, require the use of the power expansion (51) for the propagators converging at ε α , while separating the states degenerate with α in the spectral representation (14) for those propagators diverging at ε α .
So, we have a term arising from the states l non-degenerate with α S (4) plus a term derived from states α degenerate with α, S with the non divergent (at ε α ) propagators evaluated at ε α − k 0 in both cases.On the other hand, taking the derivative term (51) from one propagator and evaluating the other at ε α − k 0 it gives Finally, there is also a term arising from the derivative term in both propagators.However, that term becomes of the order O( 0 ), so it does not give any level shift and can be disregarded.The S (4) V R contribution can be split similarly, too.
In the treatment of the terms S LL and S (4) RR , we separate the degenerate states α and α from the spectral representation of the two propagators which diverge at ε α , and expand the other one according to (51).We shall not give here the details of the resulting terms which are rather involved; but, as an illustration, we can consider, for example, those parts of the propagators depending on the states l and n having energies and similar terms for S

Discussion
We derived in the previous section the various terms of the S−matrix contributing to the energy shifts given by Eq. ( 25).It should be noted that some of them (e.g., S V V , S LV non−deg or S LV deg .These terms diverge as approaches zero.The same happens with those terms of Eq. (25) containing products, like S (1) S (3) .In fact, those terms cancel out the −divergences from the ones coming from S (4) (1/ n ) such that n > 1.Hence, finally, we are left with energy expressions which does not diverge at the limit → 0.
In order to gain insight about the final expressions, let us discuss the energy contributions from the terms S and The last two formulas are relatively simple and easy to interpret.∆E LV can be thought as a second order perturbation theory, written in terms of orbital energies and one-electron matrix elements, between the perturbation U and the self-energy corrected perturbation Λ.However, Λ can not be given in a closed form and it is only defined through its matrix elements already defined in Eq. (48).From its definition, we see that Λ lα contains the external potential V = γ 0 U such that the whole expression ∆E LV becomes quadratic in U , as we expected.Furthermore Λ lα has contributions from the complete spectrum of photon frequencies k 0 .On the other hand, the energy ∆E LL shows explicitly its quadratic dependence on V , and it has a form resembling third order perturbation theory with the self-energy insertion Σ nα as an additional perturbation.Both ∆E LV and ∆E LL are feasible to be implemented in currently existing computational codes.
Finally, the two-vertex contribution ∆E V V cannot be written as a perturbation theory-like expression, showing its intrinsic QED origin.Terms like this cannot be derived from standard perturbative methods, and illustrate the usefulness of the theory presented in this work.

Concluding Remarks and Perspectives
We have given in this paper a theory for the inclusion of self-energy corrections to the nuclear magnetic parameters J and σ.It is based on the S−matrix formulation of bound-states quantum electrodynamics.Explicit expressions were given for first up to fourth order S−matrix terms.Divergent terms at the limit → 0 cancel out each other and −independent energy contributions are obtained.The resulting expressions have standard perturbation theory forms with new QEDderived perturbation operators added.
Some work remains to be done to get formulas computationally implementable in widely used relativistic electronic structure computational codes.Particularly, a regularization scheme must be applied in order to render our final formulas ultraviolet and infrared convergent.In previous works, dimensional regularization have proven to be effective.It should also be noted that we have not considered here other QED radiative effects like vacuum polarization, which are expected to be of the same order than those included in this work.

Figure 2 :
Figure 2: Feynman diagrams side left (L), side right (R), and vertex (V) for self-energy correction corresponding to the S (3) λ term.

8
however, contain one fermion propagator diverging at the relevant point ε α .Finally, the contributions S RR have two divergent propagators at the occupied orbital energies.Then, the term S

4 )
LV non−deg and S (LL non−deg .The level shift formula Eq. (25) gives ∆E V V = −e 2 occ α d 3 w d 3 x d 3 y d 3 z dk 0 11. ..H I (x k ), and ...c represents connected diagrams corresponding to expectation values on the state |0 and it will be implicitly understood in the following.Wick's theorem11allows to express a T −product of any set of Fock operators A, B, C, . . .X, Y, Z in terms of their normal ordered product and one, two, etc. contractions, T {ABC . . .XY Z} = : ABC . . .XY Z : + : AB C . . . XYZ : + : ABC . . .XY Z : + . . .