Gauge-invariant expectation values of the energy of a molecule in an electromagnetic field

In this work, we partition the full Hamiltonian for a molecule in a time-dependent electromagnetic field into a molecular Hamiltonian and a field Hamiltonian, both with gauge-invariant expectation values. The Hamiltonian for the molecule plus the electromagnetic field yields the total energy, which is gauge-independent.¹ In contrast, the most common partitioning of this Hamiltonian introduces gauge dependence into both the molecular Hamiltonian and the field Hamiltonian. The molecular Hamiltonian gains an extra term that depends on the gauge potential φ_Λ(r, t), as in the work of Kramers,² Cohen-Tannoudji, Diu, and Laloë,³ Haller,⁴ Scully and Zubairy,⁵ Sauer,⁶ and others. The inclusion of the gauge potential preserves the form of the time-dependent Schrödinger equation for a molecule in an applied field upon gauge transformation, provided that the wave function is multiplied by the appropriate gauge-dependent phase factor. However, the gauge potential serves solely to cancel a term in the time-dependent Schrödinger equation that arises from the variation of the non-physical, gauge-dependent phase factor with time. There is no contribution to the underlying dynamics of the quantum system.

The gauge potential alters both the expectation values of the molecular Hamiltonian and the differences between the expectation values for different wave functions. For this reason, it has been claimed that the Hamiltonian does not represent a true physical quantity.^2,3

In this work, we provide an alternative partitioning of the full Hamiltonian into a molecular term H_m and a field term H_f, which yields gauge-independent expectation values for both. The expectation value of H_m gives a physically meaningful result for the energy of a molecule in an external electromagnetic field. Gauge dependence can be eliminated from the molecular Hamiltonian because the apparent contribution of the gauge potential to the energy of the molecule cancels exactly with a gauge-dependent term in the usual Hamiltonian for the field. Assigning both gauge-dependent terms to the same subsystem—either to the molecule or to the field—yields gauge-independent expectation values for both subsystems.

For an isolated molecule in the absence of an external field, the full Hamiltonian includes the energy of the electromagnetic field that is generated by the electrons and nuclei in the molecule. In the Coulomb gauge, the potential for interactions of the electrons and the nuclei is the instantaneous Coulomb potential. In typical ab initio electronic structure calculations, the vector potential due to the electronic and nuclear currents is neglected in the absence of an applied field (cf. Ref. 7), and physical meaning is ascribed to the transition energies derived from the resulting molecular Hamiltonian.

When there is no external electromagnetic field, the external vector and scalar potentials are usually set to zero as well; but it is possible to perform a gauge transformation that sets up non-zero external potentials, while the external electromagnetic field remains zero.^8–10 By using the same analysis that has customarily been used for non-zero fields,^2–6 we show that this zero-field to zero-field gauge transformation could disrupt the expectation values of virtually any Hamiltonian, by introducing a gauge potential. This would make the molecular Hamiltonian in zero applied field a non-physical operator—clearly a problematic result. Our repartitioning of the Hamiltonian eliminates this problem.

Renewed interest in the effects of gauge transformations on the Hamiltonian has been raised in particle physics, due to questions about the separation of spin and orbital angular momentum of particles in gauge theories.¹¹ Forms of the Hamiltonian that are completely gauge-independent have been suggested^11–13 and applied in work on the gluon spin,^12–14 momentum,^15,16 and polarization,^17,18 nucleon spin and orbital angular momentum,^19–21 and also in work on the time-dependent harmonic oscillator²² and the Dirac equation for the hydrogen atom.²³ 

When a time-dependent electromagnetic field with a vector potential A(r, t) and scalar potential φ(r, t) acts on a system of charged particles, the kinetic energy operator for a particle (indexed by α) with mass m_α is given by [p_α − q_α A(r_α, t)]²/(2m_α), where p_α denotes (ħ/i) ∇_α, q_α is the charge on particle α, and r_α is its position. In the formulation of a gauge-independent Hamiltonian by Chen et al.,^11–13 the kinetic energy operator [p_α − q_α A(r_α, t)]²/(2m_α) is replaced by [p_α − q_α A_⊥(r_α, t)]²/(2m_α), where A_⊥(r_α, t) is the transverse component of the vector potential, and the scalar potential of the external field is dropped. Thus, the Hamiltonian becomes identical to the Hamiltonian in the Coulomb gauge (commentary is provided in Refs. 24–30).

Our analysis differs, in that we do not require the molecular Hamiltonian itself to be gauge-invariant, but we obtain gauge-independent expectation values of the molecular Hamiltonian, consistent with the requirements of Leader and Lorcé.^29,30 This is possible because a gauge transformation introduces a gauge-dependent phase factor into the wave function,^3,6 which compensates for the gauge dependence of the Hamiltonian when the expectation values are computed.

In previous work, we have shown that the energy of a molecule subject to a time-dependent perturbation separates exactly into an adiabatic term and a non-adiabatic term.³¹ The adiabatic term gives the energy changes due to the adjustment of the initial state of the molecule to the perturbation without transitions, while the non-adiabatic term gives the energy change due to transitions. For a molecule in an applied electromagnetic field, we have also shown that the time derivative of the non-adiabatic term in the energy is identical to the power absorbed by the molecule, calculated as the scalar product of the current density with the electric field, integrated over space.³² We carried out this analysis in the Coulomb gauge;³² the current work shows that the result holds in any gauge.

The gauge issue analyzed here differs from the gauge dependence of the Hamiltonian in calculations of static magnetic properties. For a molecule in a uniform, static magnetic field B, with φ_e(r) = 0, the vector potential A_e(r) depends on the gauge origin G in the form A(r) = (1/2) B × (r − G), and therefore the Hamiltonian depends on G. (If the gauge origin is identical to the origin of the coordinate system, then G = 0, but this need not be the case.) However, the expectation values of the Hamiltonian are not gauge-dependent in this case, if the proper phase factor is included in the wave function. The use of gauge-including atomic orbitals solves this problem (e.g., see Refs. 33–52 and the related methods in Refs. 53–64).

The gauge issue analyzed here also differs from gauge issues that may arise when transition probabilities are computed in the dipole-length, velocity, or acceleration gauges.^65–83 The interaction of a small molecule with electromagnetic radiation may be approximated at the electric dipole level by the perturbation term H′ = − E(r, t) ⋅ μ, where E(r, t) is the applied electric field and μ is the molecular dipole operator. For field-induced electronic transitions, the transition matrix elements of μ between an initial state |i〉 and final state |f〉 may be re-expressed in terms of the state-to-state transition energies, and matrix elements either of the momentum or the force acting on the electrons, by using commutators that involve the unperturbed Hamiltonian H₀ in the Coulomb gauge, with the vector potential due to the electronic and nuclear currents neglected (e.g., see Ref. 75)

and

where the sum over j runs over electrons and the sum over N runs over nuclei; r_jN denotes the vector from nucleus N to electron j, and r_jN is the length of this vector. The transition probabilities in the dipole-length, velocity, and acceleration (force) gauges are identical for the exact solutions of the Schrödinger equation, though they may differ for approximate solutions. In contrast, the expectation values of the standard molecular Hamiltonian (including the gauge potential) differ in different gauges, even when the solution is exact.

In Sec. II of this work, we start from the Lagrangian for a set of charged particles in the absence of an applied electromagnetic field. The potentials generated by the particles themselves are included in the Lagrangian and treated classically. The Lagrangian depends on the positions and velocities of the particles, and on the potentials and their time derivatives. We consider gauge transformations from zero applied field and zero external potentials to zero applied field, but non-zero external potentials.^8–10 A complication then arises in generating the corresponding Hamiltonian, because the momentum conjugate to the scalar potential vanishes identically. We resolve this problem with a method originally due to Dirac.⁸⁴ This same method was used by Goldman in converting the Dirac Hamiltonian into a generalized Pauli Hamiltonian.⁸⁵ The constraint on the momentum conjugate to the scalar potential is ensured via a Lagrange multiplier. We partition the full Hamiltonian into a molecular term and an “external field” term, both with gauge-invariant expectation values. Subsequent conversion of the positions and momenta of the electrons and nuclei into operators yields the quantum molecular Hamiltonian, with the static Coulomb potential for the interactions between the electrons and the nuclei.

In Sec. III, we perform the analogous transformation and partitioning to obtain the molecular Hamiltonian for a set of charged particles in an applied field that is non-zero. This Hamiltonian has gauge-independent expectation values as well. The molecular Hamiltonian that we obtain is equivalent to the energy operator that Yang and Kobe^86–95 obtained by requiring form invariance of the operator under gauge transformations (rather than by analyzing the field term in the total Hamiltonian, as in this work). The time-derivative of the expectation value of the energy operator gives the power absorbed by the system.^86–95 The current work, combined with the results of Mandal and Hunt,^31,32 shows that the power absorption is specifically connected to the non-adiabatic term in the energy, in any gauge.

In Sec. IV, we consider the effect of changes of gauge on the time dependence of the molecular wave function, using our partitioning. In Sec. V, we briefly summarize the most important results of this work.

In the absence of an applied electromagnetic field, the Hamiltonian for an isolated molecule written in the Coulomb gauge is the sum of the kinetic energy of the electrons and nuclei, the instantaneous Coulomb energy of their interaction, and the energy of the transverse electromagnetic fields generated by the molecule,⁹⁶ 

where p_α is the momentum of particle α, q_α is its charge, m_α is its mass, and A(r_α, t) is the vector potential, which is purely transverse in the Coulomb gauge; V_C is the operator for the Coulomb energy, computed with the instantaneous charge density operator $ρˆ(r,t)$⁠,

where ε₀ is the dielectric permittivity of free space, E_⊥(r, t) is the transverse component of the electric field due to the charged particles,⁹⁷ c is the speed of light, and B(r, t) is the magnetic field due to the particles. The vector potential A(r, t) is transverse, and the scalar potential is the Coulomb potential φ_C, computed with the instantaneous charge density. There are no applied (external) fields in this case.

Here, we perform a gauge transformation from zero applied fields E_e(r, t) and B_e(r, t) to zero applied fields^8–10 and derive the gauge-transformed Hamiltonian. Initially, the external vector and scalar potentials vanish: A_e(r, t) = 0 and φ_e(r, t) = 0. A gauge transformation yields non-zero gauge potentials A_Λ(r, t) and φ_Λ(r, t), which are connected to an arbitrary differentiable function Λ(r, t) by

The external fields E_e(r, t) and B_e(r, t) remain zero under this transformation,⁹⁸ since

and

After the gauge transformation, the vector potential is the sum of the (external) vector gauge potential A_Λ(r, t) and the transverse vector potential A_m(r, t) that is generated by the molecular charges. The total scalar potential is the sum of the scalar gauge potential φ_Λ(r, t) and the Coulomb potential φ_C(r, t).

To construct the gauge-transformed Hamiltonian, we start with the Lagrangian for the molecule coupled to the electromagnetic field. The Lagrangian is treated as a function of the variables {r_α}, {dr_α/dt}, and {A(r, t), ∂A(r, t)/∂t, φ(r, t), ∂φ(r, t)/∂t} at each point in space r. Here, A(r, t) is total vector potential and φ(r, t) is the total scalar potential. Then⁹⁹ 

The canonical momentum conjugate to a coordinate [{r_α} , A(r, t), or φ(r, t) for each r value] is obtained as the derivative of the Lagrangian with respect to the time-derivative of the coordinate. For the particles,

From Eqs. (7) and (9), the momenta corresponding to the Cartesian components A_i(r, t) of the vector potential A(r, t) are

for i = 1, 2, 3, corresponding to x, y, and z. In Eq. (11), E_i(r, t) is a Cartesian component of the total electric field. The momentum π₀(r, t) corresponding to the scalar potential φ(r, t) vanishes, since

This result interferes with the standard construction of the Hamiltonian.^{84,85,100,101} Dirac proposed a method of including π₀(r, t) in the Hamiltonian by use of a Lagrange multiplier.⁸⁴ This permits the construction of the full Hamiltonian, using

The inclusion of π₀(r, t) in the Hamiltonian is unrelated to the separation of H into field and molecular terms, which is analyzed in this work. Goldman⁸⁵ has used the same method of including π₀ in the Hamiltonian, while making a Foldy-Wouthuysen transformation^102–104 from the Dirac Hamiltonian to the generalized Pauli Hamiltonian. In this work, we have employed the method^84,85 in a nonrelativistic context. In the  Appendix, we recast the Lagrangian of Eq. (9) using a Lagrange multiplier, and then analyze the Hamiltonian derived from Eq. (13) to obtain

Both π₀(r, t) and its time-derivative vanish by Hamilton’s equations of motion; thus the final two terms in Eq. (14) vanish. Therefore, written in terms of the charge-density operator $ρˆ(r,t)$⁠, the Hamiltonian of Eq. (14) is equivalent to

By applying Gauss’s theorem to the final term in the Hamiltonian of Eq. (15) (cf. Ref. 85 for the relativistic version), for a neutral molecule in the absence of an applied field, we transform the total Hamiltonian H to

In this case, the only charges in the system are the charges of the nuclei and the electrons in the molecule. From the Maxwell equation for [∇ ⋅ E(r, t)] (see also Ref. 85),

where $〈ρˆ(r,t)〉$ is the expectation value of the charge density of the molecule. Therefore,

Because of the cancellation shown in Eq. (18), the only remaining gauge-dependence in H comes from the kinetic energy term [p_α − q_αA(r_α)]²/(2m_α), for each of the particles α: the gauge transformation adds a longitudinal component to A(r, t). In addition, the wave function gains a gauge-dependent phase factor as a result of the gauge transformation of the vector and scalar potentials in Eqs. (5) and (6),^3,6

When the phase factor in the wave function is taken into account, the addition of the longitudinal component ∇Λ (r, t) to A(r, t) has no effect on the expectation value of the Hamiltonian, as shown explicitly in the supplementary material¹⁰⁵ (see also Ref. 6).

In effect, the standard analysis separates the total Hamiltonian into two parts, H_ms for the molecule and H_fs for the electromagnetic field, both of which are gauge-dependent,

A gauge transformation from zero applied field to zero applied field changes the expectation value of H_ms by $〈ψ|−∫drρˆ(r,t)∂Λ(r,t)/∂t|ψ〉$⁠. This matrix element is non-zero in general, and it is generally different for different eigenstates of the unperturbed Hamiltonian H₀. Therefore, if the standard molecular Hamiltonian is used, the transition energies between states are disrupted by gauge transformations, even in the absence of an applied field. That is, the transition energies for any physical system could be altered by a gauge transformation from zero field to zero field. We note that E_⊥(r, t) and B(r, t) in the field Hamiltonian H_fs are non-zero, because the charges and currents in the molecule generate electromagnetic fields; only the external field vanishes in the present case.

A gauge function that satisfies the wave equation without introducing any non-physical charge densities or currents¹⁰⁶ is

where C is a constant and k = ω/c. The corresponding gauge potential is

For the hydrogen atom in the 1s state, the expectation value of the gauge potential in Eq. (23) is

where a₀ is the Bohr radius. For comparison, for the hydrogen atom in the 2s state, the expectation value of the gauge potential is

Thus, the transition energy between the 1s and 2s states is altered by the gauge transformation from zero field to zero field, following the standard analysis. For example, with the expectation values of the gauge potential in the 1s state written as Cωf_1s(k)exp(−iωt), and the expectation values in the 2s state written as Cωf_2s(k) exp(−iω t), we find f_1s(k) values of (16/25), (1/4), (16/169), (1/25), and (16/841) for k = 1, 2, 3, 4, and 5, respectively, while f_2s(k) equals 0, (21/625), (17/1250), (465/83 521), and (147/57 122) for k = 1, 2, 3, 4, and 5, respectively (with all values in atomic units). This type of apparent gauge dependence is expected, in general, for any molecule, even in the absence of an applied field. We regard this gauge dependence as an artifact of the standard separation of the Hamiltonian into H_ms and H_fs.

If the cancelling quantities in H_ms and H_fs [i.e., the last term in Eq. (20) and the last term in Eq. (21)] are grouped together, either into a modified version of the molecular Hamiltonian or into a modified version of the field Hamiltonian H_f, then the gauge potential φ_Λ(r, t) does not contribute to the expectation values of either H_m or H_f, due to Eqs. (6) and (18). As a result, both H_m and H_f have gauge-independent expectation values. The molecular Hamiltonian is not gauge-independent per se because it contains the vector potential A(r, t), but its expectation values are gauge-independent.¹⁰⁵ In Sec. III, we show that the apparent gauge dependence of the Hamiltonian for a molecule in non-zero external electromagnetic field has the same origin as in the zero-field case and can be eliminated by the same partitioning of the full Hamiltonian.

In this section, we consider a molecule acted upon by a propagating electromagnetic wave that has no charges or currents within the volume under consideration; the only charges and currents are those of the molecule. The propagating electromagnetic wave is transverse. The Lagrangian for the system of the molecule and the applied radiation field is identical to the Lagrangian in Eq. (9).

Thus for a molecule in an external electromagnetic field, the Lagrangian L has the form

The vector potential due to the sources in the molecule is denoted by A_m(r, t) and the instantaneous Coulomb potential is denoted by φ_C(r, t). The choice of the Coulomb gauge for the molecule does not constrain the gauge choice for the external field. The total vector potential is the sum of the molecular term and the external-field term, A(r, t) = A_m(r, t) + A_e(r, t); and similarly for the scalar potential, φ(r, t) = φ_C(r, t) + φ_e(r, t).

The vector potential of the external field A_e(r, t) may have a longitudinal component, if the external field is not treated in the Coulomb gauge. In that case, the longitudinal term in the electric field generated by −∂A_e(r, t)/∂t is cancelled by a longitudinal contribution to E(r, t) from the scalar potential of the external field φ_e(r, t).

An arbitrary gauge transformation converts the original vector potential A(r, t) and scalar potential φ(r, t) to the new potentials A_Λ(r, t) and φ_Λ(r, t) via

This does not change the fields E(r, t) and B(r, t). The momentum of the particle indexed by α is given by Eq. (10), and the momentum conjugate to A(r, t) is given by Eq. (11), as −ε₀ E(r, t). As before, the momentum conjugate to φ(r, t) vanishes. We modify L by the introduction of a Lagrange multiplier, as in Eq. (A1).^84,85 This permits the formal construction of the Hamiltonian, with the form in Eq. (13). The analysis of Eq. (13) given in the  Appendix needs only a few modifications to allow for a non-zero applied field. In Eqs. (A8), (A9), (A10), (A12), and (A20), ∂∇Λ(r, t)/∂t is replaced by ∂A_||(r, t)/∂t, which is the time derivative of the longitudinal component of the external vector potential. In Eqs. (A13) and (A20), φ_e(r_α, t) replaces −∂Λ(r_α, t)/∂t. This converts the Hamiltonian to the form

As before, the momentum π₀(r, t) conjugate to the scalar potential vanishes. The term in Eq. (29) containing the scalar potential of the external field is rewritten in terms of the charge density operator as

The longitudinal component E_||(r, t) of the electric field of the applied radiation is given by

and since the external radiation field is transverse, E_||(r, t) = 0. Therefore,

Then an application of Gauss’s theorem and the Maxwell equation for the divergence of E(r, t) in Eq. (17) gives

That is, the expectation value of the term in Eq. (30) cancels identically with the term in Eq. (32). If both terms are assigned either to the molecular Hamiltonian or to the field Hamiltonian, then both will have gauge-independent expectation values; and these will be identical to the expectation values of the molecular Hamiltonian H_m given by

and the field Hamiltonian H_f given by

Only the transverse electric field appears in H_f, because the energy of the longitudinal electric field is given by the Coulomb energy V_C, and the external radiation field is transverse. The expectation values of H_m are identical to those computed directly in the Coulomb gauge.

In this section, we consider the effect on the time-dependent Schrödinger equation of a gauge transformation from the original vector potential A(r, t) and scalar potential φ(r,  t) to the new potentials A_Λ(r,  t) and φ_Λ(r,  t), via Eqs. (27) and (28). We use ψ(r₁, r₂, …, r_n, t) to denote the solution of the Schrödinger equation with the original potentials A(r, t) and φ(r,  t). A gauge transformation introduces a phase factor into the wave function ψ_Λ(r₁, r₂, …, r_n, t), as given in Eq. (19).^3,6 That is, the gauge-transformed wave function acquires additional time dependence that does not reflect the actual dynamics of the physical system but rather arises from the variation of the gauge function with time.

Inclusion of the gauge potential in the molecular Hamiltonian (in the standard approach) counters the effects of the time-dependence of the phase factor. If the time-dependent Schrödinger equation restricted to the molecule holds in the original gauge,

then an equation of the same form also holds after gauge transformation. The Hamiltonian acting on the wave function converts in the standard analysis to

The time derivative of the gauge-transformed wave function satisfies

The intrinsic dynamical evolution of a molecule in an electromagnetic field is independent of the gauge potential. The longitudinal terms ∇Λ(r_α, t) in the vector potential do not affect the expectation values of the Hamiltonian,^6,105 and

The scalar potential term in the standard molecular Hamiltonian serves solely to compensate for the time dependence of the gauge potential, since

The terms on both sides of Eq. (40) vanish in the Coulomb gauge, because the scalar potential of the external electromagnetic field vanishes in that gauge.

We note that when the momentum operator for a particle is modified from (ħ/i)∇_α to [(ħ/i)∇_α − q_α A_e(r_α, t)], then in order to form a fully covariant derivative vector, the time-derivative operator must also be modified to [−(ħ/i)∂/∂t − q_α φ_e(r_α, t)]; this operator is form-invariant under gauge transformations, as well.⁸⁸ With this modification of the time-derivative operator, the form of the time-dependent Schrödinger equation is preserved upon gauge transformation, with our partitioning of the full Hamiltonian into H_m and H_f in Eqs. (34) and (35).

The full Hamiltonian for a molecule in an electromagnetic field, including the field energy, is invariant under gauge transformations.¹ The standard partitioning of the full Hamiltonian results in a molecular Hamiltonian that contains a gauge potential,^2–6 which leads to gauge dependence of the expectation values. For this reason, it has been argued that the Hamiltonian is not a physical operator.^2,3 In Sec. II, we have shown that a gauge transformation from zero applied field to zero applied field, when coupled with the standard partitioning of the Hamiltonian, would render the expectation values of virtually every molecular Hamiltonian and the transition energies between states gauge-dependent, and hence non-physical. We provided an explicit example of this gauge dependence for a hydrogen atom, in the absence of an external field.

We have resolved this issue of gauge dependence by identifying an alternative partitioning of the full Hamiltonian into a molecular term H_m given in Eq. (34) and a field term H_f given in Eq. (35). Both have gauge-independent expectation values. The partitioning is applicable for zero or non-zero applied field.

As the principal result of this work, we have shown that the expectation values of the molecular Hamiltonian H_m give the energy of a molecule in a time-dependent electromagnetic field, in an unambiguous, gauge-independent form. Any gauge may be selected for calculations, with the same results after the partitioning. The expectation values of the partitioned molecular Hamiltonian H_m are identical to those obtained directly in the Coulomb gauge.

As a corollary of this work and our earlier results in Ref. 31 and 32, it follows that the power absorbed by a molecule from an external electromagnetic field is equal to the time-derivative of the non-adiabatic term in the energy, in any gauge.

This work has been supported in part by NSF Grant No. CHE-1300063 from the Chemical Theory, Models and Computational Methods Program in the Chemistry Division of the NSF.

Following the method suggested by Dirac,⁸⁴ we add a Lagrange-multiplier term to the Lagrangian from Eq. (9) to obtain

where κ is the Lagrange multiplier, an arbitrary constant. The function f(r, t) is determined by the (arbitrary) choice of gauge; and κ is used to impose the gauge constraint

The momentum conjugate to φ(r, t) is then

The momenta corresponding to the Cartesian components of the vector potential are unchanged by the addition of the last term in Eq. (A1) to L.

From Eq. (13) for the Hamiltonian with Eq. (A1) for the Lagrangian, we have

The terms that depend explicitly on (dr_α/dt) simplify to give the kinetic energy, when summed over the particles indexed by α. From Eq. (10),

and so

From Eq. (11), the part of the second term in (A4) that contains the time derivative of the vector potential transforms as follows:

Then we observe that

So

When integrated over all space, the longitudinal and transverse electric fields are orthogonal. Therefore,

This result combines with the part of the fourth term in Eq. (A4) that depends on E²(r, t), i.e.,

So the quantity on the left in Eq. (A10) combines with the quantity on the left in Eq. (A11) to give

The term in Eq. (A12) that contains the square of the longitudinal field cancels in part with the fifth term in Eq. (A4), which contains the charges and the scalar potential. The fifth term is

where V_C is given by Eq. (4) of the main text. The only longitudinal field is the instantaneous Coulomb field, so E_||(r, t) is related to the Coulomb potential φ_C(r, t) by

Using Maxwell’s equation for the divergence of the electric field [Eq. (17) of the main text] and Gauss’s theorem, we obtain

Since ∇ × A(r, t) = B(r, t), the part of the fourth term in (A4) that depends on the curl of A(r, t) has an equivalent expression in terms of the magnetic field B(r, t),

The terms in Eq. (A4) that have not yet been analyzed contain π₀(r, t), either explicitly or by use of Eq. (A3). These terms simplify as follows:

From (A3),

So

Combining the results from Eqs. (A6), (A12), (A15), (A16), and (A19), we obtain

Eq. (A20) is identical to Eq. (14) of the main text.