# Symmetries, Conserved Properties, Tensor Representations, and Irreducible Forms in Molecular Quantum Electrodynamics

## Abstract

**:**

## 1. Introduction

_{2v}as opposed to D

_{∞h}) that H

_{2}O possesses an electric dipole moment—and every life as we know it could not exist otherwise.

## 2. Charge-Parity-Time Symmetry in Molecular Electrodynamics

_{2}, whose ±1 eigenvalues signify even or odd parity. All optical phenomena preserve symmetry under the product operation $\mathcal{C}$$\mathcal{P}$$\mathcal{T}$—a proof of this universality and analysis of its implications has been authoritatively presented in a recent review by Lehnert [12], and a broad spectroscopic perspective on the topic has been given by Lazzeretti [14]. Nonetheless, considerations of charge conjugation symmetry are seldom relevant for conventional electrodynamic phenomena, as the mathematical operation $\mathcal{C}$ is never physically realized; clouds of negative charge always surround positively charged nuclei. Accordingly, in the consideration of optical effects, it is usually sufficient to restrict consideration to the $\mathcal{P}$$\mathcal{T}$ product, which, through the constraints of Lorentz invariance, ensures Hamiltonian operators of Hermitian form. Moreover, $\mathcal{P}$$\mathcal{T}$-symmetric quantum theory has been shown to be exactly equivalent to standard (Hermitian) quantum mechanics in terms of all observables [15].

## 3. Dual Symmetry and Conservation Laws in Quantum Electromagnetism

**e**is formally of odd parity under $\mathcal{P}$ as well as under $\mathcal{T}$; the converse applies to the magnetic induction field

**b**. This symmetry is indeed required by the structure of the Faraday and Ampère Laws. Nonetheless, these and the other two Maxwell’s equations also support another well-known, fundamental symmetry, registering a dual complementarity between the electric and magnetic fields of optical radiation in free space. It is a symmetry that is compromised in the presence of electric charge, owing to the asymmetry in existence of electric but not magnetic monopoles; for the electric field, a charge-driven source term accordingly appears in Gauss’s Law, but there is no counterpart in the expression for divergence of the magnetic field. Nonetheless, there is sufficient interest and power in the underlying free-space symmetry that there is recurrent attention in electromagnetic duality. Indeed, much of the recent interest—largely centred on structured and singular light, with associated momentum and angular momentum issues—does concern essentially free-space propagation.

**r**and t are space and time coordinates. For more general application, it is the transverse electric displacement field ${\mathit{d}}^{\perp}$ that should feature in (1), rather than the electric field

**e**, but in source- and current-free regions, there is no physical distinction (the symbol

**d**is also commonly used in entirely different connections). Here, too, with a view to the microscopic formulation that is appropriate for application to systems on the molecular scale, the standard lower-case symbols are used; the context will generally make it clear if quantum operators are signified. Notably, in the above Equation (1), symmetrisation is necessary to ensure Hermiticity, because of the non-commutativity of the electric and magnetic field operators at a common point in space [4,22]:

**e**and

**b**is manifested in the form of the duality transformation under which Maxwell’s source-free equations prove invariant:

**e**and

**b**fields exhibit different spatial and temporal parities because they have different physical dimensions.

**j**into spin and orbital parts,

**s**and

**l**, respectively, proceeds along the following lines [25]:

**a**is the vector potential field. Quite apart from the gauge-dependence of

**a**, it is well known that this separation is beset with problems; the spin operator

**s**does not satisfy the necessary commutation relations amongst its Cartesian components, to be acceptable as a true quantum mechanical operator [26]. As pointed out by Barnett et al., the same conclusion therefore necessarily applies to the counterpart orbital angular momentum

**l**, as the sum of the two does constitute a mathematically correct formulation of the orbital momentum from the vector product $\mathit{r}\times \mathit{p}$. [27]. Their work nonetheless exhibits the dual transformation as essentially consistent, within the paraxial approximation, to the rotations generated by treating

**l**and

**s**as infinitesimal angle generators.

**f**[36,37,38]:

**k**and polarization η; there is no way to represent the wavefunction for the two-photon state, $|2(\mathbf{k},\eta )\rangle $, as any kind of combination or product of one-photon $|1(\mathbf{k},\eta )\rangle $ state functions (just as it is not possible to represent the wavefunction for a 2s electron in hydrogen in simple terms of 1s wavefunctions). The notion of a photon wavefunction can serve as a workable pragmatism when single photons are involved, and the distinction from a state vector poses less of a problem, but for states with two or more identical photons, there is no conventional sense in which any one photon can be considered to have its own wavefunction [42].

**a.b**emerges as follows:

**φ**is a polar vector field, even under $\mathcal{P}$ and odd under $\mathcal{T}$. Together, the operators defined by Equations (11) and (12) represent components of a four-vector $\left(c\chi ,\mathit{\phi}\right)$ in Minkowski space [49], signifying the conserved Lipkin ‘zilch’ [43].

## 4. Symmetry Principles for Photon–Molecule Interactions

_{FI}, the matrix element of an operator M that couples an initial state $|I\rangle $ to a final state $|F\rangle $ in a composite system (i.e., molecule plus radiation). In the present connection, with a focus on processes in which energy is exchanged between the radiation and matter, the final state is presumed to be measurably different from, though necessarily isoenergetic with, the initial state of energy ${E}_{I}$. The operator M may itself be cast in the following resolvent operator form [22]:

_{0}is the basis Hamiltonian, comprising the unperturbed molecular and radiation operators. Implementing the completeness relation delivers the system matrix element ${\left({M}_{FI}\right)}_{\mathrm{sys}}$ in the form of a familiar expansion in the light-matter interaction operator H

_{int}, representing a time-dependent perturbation:

_{R}, E

_{S}, and so on, are also cast in the system basis. Each Dirac bracket featured in the numerators of terms in Equation (17), and thus entails both matter and radiation components—and to identify symmetry aspects, both must be brought into explicit consideration.

_{0}is separable in each component. Clearly, all energies are scalar quantities, and H

_{mol}is invariant under the same full set of symmetry operations as the molecule, whose symmetry class is always identified with the ground state (or higher, in the case of chiral species [60]—where the ground state wavefunction lacks a two-fold permutational symmetry that is present in the molecular Hamiltonian).

**b**. This affords major calculational advantages and insights; expressing the couplings between the optical fields and charges directly in terms of experimentally meaningful electric and magnetic fields also highlights their involvement with corresponding multipole moments and optical response tensors in Cartesian form, thus elucidating their connection to molecular symmetry. Strictly, when casting theory in terms of a PZW Hamiltonian formulation, all orders of the electric multipole coupling should be cast in terms of a transverse electric displacement ${\mathit{d}}^{\perp}$, rather than the electric field ${\mathit{e}}^{\perp}$ [4,61]. However, in isotropic media such as gases, and all conventional liquids and solutions, the two quantities are related by a scalar, so precisely the same symmetry arguments apply. The equations here are written in terms of the electric field for simplicity, and consistency with previous work. The leading terms of H

_{int}are thus expressible as follows:

**μ**is the electric dipole operator, Q is the (second rank tensor) electric quadrupole operator, and

**m**is the magnetic dipole operator. The first and third of these are vectors; the quadrupole operator is a second rank tensor; and the indices i, j represent coordinates in any consistent frame of spatial reference with orthonormal axes—usually Cartesian, but not necessarily so (see Section 11). Every index that is repeated signifies an implied summation over the 3D basis set.

**r**, within an arbitrary quantization volume V, in terms of sums over wave-vector

**k**and polarization state η. The latter sum may in principle be taken on a basis comprising any two states that are represented by opposing points on the Poincaré sphere; [66] commonly, those chosen are either left and right circular polarizations, or horizontal and vertical plane polarizations. The circularly polarized basis can in fact be expressed in terms of the following unit vectors:

**f**annihilates a left-handed photon and creates a right-handed photon, whereas its Hermitian conjugate ${\mathit{f}}^{\u2020}$ does the opposite [37]. An important corollary follows; noting the linearity of the electromagnetic fields in H

_{int}(a feature that also carries through to the RS expression of coupling, see below), it becomes evident from the above sequence of expressions that the n

^{th}term in the matrix element M

_{FI}, Equation (17), delivers the leading contribution for any process involving n photons.

_{int}is an energy operator, and therefore even in both space and time, the electric dipole operator

**μ**is necessarily also odd with respect to parity $\mathcal{P}$, and even with respect to $\mathcal{T}$, its magnetic counterpart

**m**is even in $\mathcal{P}$ and odd in $\mathcal{T}$. Accounting for the gradient operator featured in the second term of (18), the electric quadrupole operator

**Q**has to be even in both forms of parity. To illustrate the significance of a difference in spatial parity, it emerges that the difference between electric and magnetic transition moments is the key to most common forms of chirality-sensitive response. As the former are polar vectors (odd in $\mathcal{P}$), and the latter are axial vectors (even in $\mathcal{P}$), it takes a molecule with no center of symmetry—that is, one that is not invariant under $\mathcal{P}$, such as a chiral molecule—to support an electronic transition involving both electric and magnetic transition moments. It is indeed an interference of these two kinds of coupling that proves to supply the main mechanism for chiroptical differentiation—see the literature for an example [67].

**f**. So although, for chiral molecules, transition dipoles based on Equation (23) may comprise non-vanishing contributions from both its electric and magnetic components, the difference in selection rules that applies for most other materials means that

**d**itself cannot be regarded as a secure gauge of chiral propensity. Moreover, for many chiroptical processes, E2 contributions do not indeed disappear on orientational averaging; Raman optical activity is a familiar example [68,69]. Any advantage of deploying the RS formulation for light-matter coupling is therefore circumscribed; the representation is not generally applicable.

## 5. The Coupling of Radiation and Molecular Tensors

_{0}, it follows that the irrep χ(M) for M is a product of the individual irreps for each of the multipoles involved in the whole process. Attending to the leading multipole terms given in Equation (18), we can write the following:

_{n}rotations to which a ground state is invariant, excited states acquire an integer power of the phase factor exp(2πi/n). Consider, for example, each term of the matrix element for a two-interaction process (noting that more than one term will usually arise, because all sequences of interaction are accommodated in the theory). Each term may entail one Dirac bracket of E1 form and the other of M1 form; all combinations of multipoles are possible in principle, though not all will necessarily be symmetry-allowed. Nonetheless, a first step is to consider what constraints are imposed on each individual interaction, as a result of the group theoretical rules imposed by molecular symmetry [70].

**S**and a molecular response tensor

**T**. Specifically, ${\mathit{S}}^{\left(r\right)}\equiv {S}_{{i}_{1}{i}_{2}\dots {i}_{r}}$ comprises an outer product of components of the electric field and the magnetic field (and in addition, where quadrupoles are involved, the field wave-vector); the corresponding molecular tensor ${\mathit{T}}^{\left(r\right)}={T}_{{i}_{1}{i}_{2}\dots {i}_{r}}$ entails products of n Dirac brackets, and its spatial symmetry properties are determined by Equation (24). Each tensor has a rank r given by r = (e + m + 2q) so that the inner product contrasts this number of indices; the molecular tensor ${\mathit{T}}^{\left(r\right)}$ specifically incorporates (e + m + q) products of transition multipole moments.

_{FI}has the physical dimensions of energy, the ${\mathit{S}}^{\left(r\right)}$ and ${\mathit{T}}^{\left(r\right)}$ tensors must have identical signatures of parity for each separate parity operation, $\mathcal{P}$ and $\mathcal{T}$. The respective eigenvalues are (–1)

^{e}and (–1)

^{m}, as determined by the space-odd, time-even character of the electric field, and the space-even, time-odd character of the magnetic field. Any electric quadrupole, having even parity under both $\mathcal{P}$ and $\mathcal{T}$, plays no part in this determination. If, for example, the ${\mathit{S}}^{\left(r\right)}$ and ${\mathit{T}}^{\left(r\right)}$ tensors are odd with respect to both parity operations, their product will remain the same if both radiation and matter are inverted in space, physically representing opposite parity enantiomers, and also opposite helicity radiation.

_{2}O

_{2}; in its ground electronic state, it has only C

_{2}rotational symmetry and is therefore chiral in principle, but it is not normally regarded as such—because at common ambient temperatures, where the substance is a liquid, thermal energy is sufficient to provide equilibration between the two oppositely handed forms. Relatively low potential energy barriers must be surmounted for interconversion to occur [71]; in this case, evidence is readily afforded by the significant energy splitting between even and odd parity combinations of the two enantiomeric state functions [72].

_{3}, which possesses, in addition to a pure rotational (C

_{3}) axis, a plane of symmetry (it belongs to the C

_{3h}point group); it is not intrinsically chiral, but if the molecule is held at a fixed angle with respect to any transversely propagating signal beam of light, it has the capacity to differentiate between circular polarizations. This type of effect—essentially 2D chirality—is more commonly encountered (and more easily registered) in the surface features of suitably fabricated metamaterials—gammadion structures are a well-studied example—where even in the absence of an external stimulus, there is a clear disparity across the planar interface between physically dissimilar regions. In this way, effects more commonly associated with optical activity may be exhibited by an intrinsically achiral material or metamaterial [73]. Nonetheless, consideration of the complete light-matter system reveals that chiroptical differentiation will only be manifest in optical fields with a helical character—either through circular polarizations, in chirally configured beams, or within optical nanofibres [74]. When circularly polarized light impinges upon a suitably nanostructured surface, propagation by reflection or transmission may exhibit differences according to direction of travel, as opposite directions are not equivalent under the operations of spatial parity $\mathcal{P}$ [75].

## 6. Structure and Permutation Symmetry in Material and Radiation Tensors

^{3}molecular tensor, written as a sum of three corresponding terms, accounting for overall energy conservation in each case, is as follows:

_{ab}denotes an energy difference E

_{a}− E

_{b}. Three terms arise because this is the order of index permutations given by the symmetric group product S

_{3}× S

_{2}.

^{2}M1 and E1

^{2}E2. The associated ‘

**J**’ and ‘

**K**’ tensors molecular tensors retain index permutational symmetry if the M1 or E2 interaction is involved in the output emission, but not if it is linked with one of the two input photon annihilation events [80,81].

**S**tensor acquires full permutational symmetry amongst all three of its indices—and by similar arguments to those presented above, the same index symmetry is effectively conferred upon the molecular tensor.

## 7. Observables

_{FI}to distinguish expectation values (signifying identical initial and final system states) from the off-diagonal matrix elements that feature as modulus squares in process observables. The distinction, recently re-emphasized by Stokes [82], becomes especially important when physically identifiable effects arise from the interference between terms involving different kinds of multipolar coupling—chiral and mechanical effects in particular, as shown in other recent work [83,84,85] To secure an expression for the rate of an observable transition process, we now work from Equation (13) to arrive at the following:

^{2}contribution to the rate equation, $\mathsf{\Gamma}={\left|{M}_{FI}^{(\mathrm{E}1)}\right|}^{2}$, is expressible in terms of the product ${\mathit{S}}_{1;0;0}^{\left(1\right)}\otimes {\overline{\mathit{S}}}_{1;0;0}^{\left(1\right)}{\odot}^{2}{\mathit{T}}_{1;0;0}^{\left(1\right)}\otimes {\overline{\mathit{T}}}_{1;0;0}^{\left(1\right)}$. Here, the material and radiation tensor constructs, as defined above, take the form of a transition electric dipole product ${\mu}_{\lambda}^{\alpha 0}{\overline{\mu}}_{\mu}^{\alpha 0}$, and a polarization component product ${e}_{\lambda}{\overline{e}}_{\mu}$ (where these subscript indices imply components in principle referred to the molecule-fixed Cartesian frame—with implications to be addressed in the following sections). This rate contribution, which even for chiral molecules retains its sign irrespective of the enantiomeric form or the circular handedness of the input radiation, is almost invariably the term that generates the largest contribution to the absorption rate. However, attending to the terms beyond E1 in the coupling delivers a corrected rate equation of the form

## 8. Irreducible Cartesian Tensor Framework for Multiphoton Interactions

^{3}coupling also applies.

_{6}; the Schoenflies point group is O

_{h}and the odd-parity representations of weights 1, 2, and 3 are T

_{1u}, (E

_{u}+T

_{2u}), (A

_{2u}+T

_{1u}+T

_{2u}), respectively. This signifies that only vibrations of A

_{2u}, E

_{u}, T

_{1u}, or T

_{2u}symmetry can produce a hyper-Raman signal. For vibrations of all other symmetries, the process is forbidden. It is to be emphasized that the symmetry properties of the transition are key here—not the permanent properties of the molecule itself. Again, taking the instance of SF

_{6}; because it is octahedral, it has no permanent hyperpolarizability—and as such, it cannot exhibit the elastic frequency doubling process of second harmonic generation (SHG). Nonetheless, the molecule can produce a hyper-Raman spectrum.

^{j}-pole, e.g., a deviator is identified as quadrupolar. In its own specific context, where it is implicit that every photon interaction in fact has E1 form, there is no likelihood of confusion, but the potential ambiguity is noted.)

_{(λμν)}, one hidden implication was that weight 2 contributions could not arise. In the SF

_{6}case examined above, this would wrongly suggest that E

_{u}vibrations are also forbidden. The essential flaws and general inapplicability of Kleinman symmetry were in fact quickly pointed out by Wagnière [118]. Recent work on third harmonic scattering has again shown that emphatic differences arise, according to whether or not full index symmetry is assumed [119]. As a corollary to all such cases, however, it is of interest that in a specific case where all the photons involved in the interaction have identical polarization, then, for the same reasons discussed in Section 6, the results will indeed be consistent with Kleinman symmetry.

## 9. Transition Classes and Information Content

_{h}, for example, the following classes arise for any even-parity, fourth rank tensor lacking full index symmetry: (432)—T

_{2g}; (431)—T

_{1g}; (42)—E

_{g}; (40)—A

_{1g}; and (3)—A

_{2g}. As shown in the Table, the number of classes is generally diminished by any admission of tensor index symmetry. Specific processes for which classification schemes based on these principles have been introduced are hyper-Raman scattering, [84] multiphoton absorption [85,86,87,88], and third harmonic scattering [96].

## 10. Isotropic and Axial Invariants and Ensemble Averages

**П**as given by Equation (30). To most simply illustrate the implementation of an orientational average, let us restrict consideration to dipole (allowing for both E1 and M1) coupling—that is, the representation of E2 couplings, q = 0. The product tensor thus has rank e + m + e′ + m′, which equates to 2n. Again, one example from hyper-Raman scattering is the sixth rank term ${\beta}_{\lambda \left(\mu \nu \right)}^{nm}{\overline{\beta}}_{\pi \left(o\pi \rho \right)}^{nm}$.

**g**with the same, even rank (r + r′ = 2n), which comprise products of Kronecker deltas. Averaging can proceed on this basis using Equation (32) in the literature [9]—which also provides for more complicated cases—but by utilising irreducible forms, we now take a different tack. The inner product of the

**П**and

**g**tensors generates results of the following form, utilizing Equation (35) from the present section and the defining Equation (30) for the explicit form of

**П**(while the

**Σ**tensors are treated in the same way):

_{1}= j

_{2}, is a consequence of the range for the coupled weights being subject to an upper limit of 0—as the isotropic tensors are weight 0 alone, and the whole expression (which results in a scalar, i.e., a tensor of rank zero) must itself result in weight 0.

_{n}linearly independent set of parameters whose number follows from the multiplicity ${\tilde{Q}}_{n}^{j}$ of each weight, as listed in Table 1; each weight only couples with itself, and hence we have the following:

**T**); if the molecular tensor can be treated as real (which generally applies for E1 coupling in regions far from optical resonance), then it follows that the number of invariants reduces to the following:

_{n}= 15 molecular invariants. Because, in general, the radiation tensor is subject to the same development, the rate (29) in principle accommodates ${t}_{3}^{2}=$ 15

^{2}= 225 terms (for four-photon processes lacking permutational symmetry, the corresponding number is ${t}_{4}^{2}=$ 8281). However, if the molecular tensor is real, ${\stackrel{\u2322}{t}}_{3}=11$ and the number of terms in the rate is almost halved.

**Σ**, using electromagnetic fields of conventional, plane wave form to only comprise any chosen, arbitrary combination of weights. By exploiting the orbital angular momentum of structured beams, Molina-Terriza et al. have shown that it is in fact possible to prepare photons in multidimensional vector states of angular momentum [130]—but orbital angular momentum is known not to engage with the leading E1 form of coupling for electronic transitions [131]. In consequence, as observed earlier, to secure the fullest information from separate experiments with different polarization conditions, the required number of studies always exceeds the number of distinct symmetry classes. Whichever method of tensor representation is deployed (reducible or irreducible), it is noteworthy that it is unnecessary to derive expressions for individual tensor components; they are not required, nor are they measurable in fluid media.

**П**and

**Σ**with isotropic tensors of rank (2n + 1), as shown in Andrews [9] and detailed in reference Wagnière [132].

## 11. Intricate Aspects of Dichroism

**П**and

**Σ**tensors constructed according to Equations (30) and (31), each to be contracted with an isotropic tensor of the same rank that is, a product of two Kronecker deltas. In particular, the field tensor

**Σ**comprises products of components of ${\mathit{\epsilon}}^{\left(\eta \right)}\left(\mathit{k}\right)$, ${\overline{\mathit{\epsilon}}}^{\left(\eta \right)}\left(\mathit{k}\right)$,

**k**, and

**B**(one component of the polarization vector, one of its complex conjugate, one of the wave-vector, and one of the magnetic field). Therefore, the result of contraction with two deltas, which produces two scalar products, may appear to be non-zero and acquire its maximum value if the static field is aligned with the direction of beam input; because ${\mathit{\epsilon}}^{\left(\eta \right)}\left(\mathit{k}\right)\cdot {\overline{\mathit{\epsilon}}}^{\left(\eta \right)}\left(\mathit{k}\right)=1$ for any polarization, the result is ostensibly non-zero. An interesting aspect for chiral molecules is that the two opposite enantiomeric forms would appear to produce opposite E1–E2M01 contributions of opposite signs, even when linearly polarized light is deployed. However, the molecular part

**П**of this result involves two terms, each one entailing a transition magnetic dipole moment—one with the transition dipole

**m**

^{αr}and the other

**m**

^{r}

^{0}, corresponding to the interaction vertices denoted by empty blue circles in the middle and right-hand diagrams in Figure 4. With real wavefunctions, the values of these moments are imaginary, because the angular momentum operator implicit in a magnetic moment operator is itself imaginary; hence, the associated rate contribution in fact vanishes (the rate equation entails twice the real part of this imaginary cross-term contribution).

**П**and

**Σ**tensors that arise are third rank, and accordingly, each demands contraction with the isotropic tensor of rank 3, that is, the Levi–Civita tensor. For

**Σ**, comprising a product of components of ${\mathit{\epsilon}}^{\left(\eta \right)}\left(\mathit{k}\right)$, $\widehat{\mathit{k}}\times {\overline{\mathit{\epsilon}}}^{\left(\eta \right)}\left(\mathit{k}\right)$, and

**B**, this generates a vector triple product that can again be non-zero if

**B**is aligned with

**k**. In this case, the molecular part

**П**of the result again entails two terms, from the middle and right-hand diagrams in Figure 4, but now each one comprises two magnetic moments, so that the molecular part of the rate contribution is real. The result persists for both linearly and circularly polarized light; the vector triple product entails the cross-product of ${\mathit{\epsilon}}^{\left(\eta \right)}\left(\mathit{k}\right)$ with $\widehat{\mathit{k}}\times {\overline{\mathit{\epsilon}}}^{\left(\eta \right)}\left(\mathit{k}\right)$, which equals $\widehat{\mathit{k}}$ for any polarization state—which, therefore, also includes the case of unpolarized light. This distinct difference in physical significance, compared with E1–E2M01, appears not to have been noted before.

## 12. Discussion

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three topologically distinct time-ordered diagrams (time progressing upwards) for hyper-Raman scattering from an input mode

**k**into an output mode ${k}^{\prime}$: the molecule undergoes a transition $n\leftarrow m$ via two virtual intermediate states r and s.

**Figure 2.**State-sequence diagram (time progressing to the right) for hyper-Raman scattering, subsuming all three pathways exhibited in Figure 1. Here, the interactions denoted by line sequences are colour-coded to highlight the input and output modes.

**Figure 3.**Illustration of coupling weights j

_{1}and j

_{2}, in a partially inner, partially outer products of two tensors, of respective ranks n

_{1}and n

_{2}, as given by Equation (35). Assuming j

_{1}> j

_{2}, the span of weights in the product may range from j

_{1}− j

_{2}to j

_{1}+ j

_{2}, capped by an upper limit n

_{1}+ n

_{2}− 2p that is the rank of the product tensor.

**Figure 4.**Key time-ordered diagrams for engaging a static magnetic field

**B**in the absorption of a single photon of wave-vector

**k**. The diagram on the left represents the leading term, where the red circles denote E1, E2, or M1 coupling. Additional coupling with the static field (empty blue circle) engages two distinct time-orderings.

**Figure 5.**State-sequence diagram for magnetic field engagement in single-photon absorption, connectors coloured to match the time-ordered representations of Figure 4.

**Table 1.**Maximum number of independent components for the tensors ${\mathit{T}}^{\left(n\right)}$ that most commonly arise in n-photon molecular interactions, brackets embracing indices with permutational symmetry. Illustrative examples: Abs—single photon absorption; nPA—n-photon absorption (single-beam); RRE—resonance Raman effect; HR—hyper-Raman effect; HS—second harmonic scattering; SFG—sum-frequency generation; SFS—sum-frequency scattering; 4WM—four-wave mixing; OKE—optical Kerr effect; THS—third harmonic scattering; SWM—six-wave mixing.

${\mathit{T}}^{\left(\mathit{n}\right)}$ | Effect | $\mathit{N}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(0\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(1\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(2\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(3\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(4\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(5\right)}$ | ${\tilde{\mathit{Q}}}_{\mathit{n}}^{\left(6\right)}$ | ${\mathit{t}}_{\mathit{n}}$ | ${\stackrel{\u2322}{\mathit{t}}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

n = 1 | |||||||||||

${T}_{\lambda}$ | Abs | 3 | 0 | 1 | 1 | 1 | |||||

n = 2 | |||||||||||

${T}_{\lambda \mu}$ | RRE | 9 | 1 | 1 | 1 | 3 | 3 | ||||

${T}_{\left(\lambda \mu \right)}$ | 2PA | 6 | 1 | 0 | 1 | 2 | 2 | ||||

n = 3 | |||||||||||

${T}_{\lambda \mu \nu}$ | SFG/SFS | 27 | 1 | 3 | 2 | 1 | 15 | 11 | |||

${T}_{\lambda \left(\mu \nu \right)\hspace{0.17em}}$ | HR/SHS | 18 | 0 | 2 | 1 | 1 | 6 | 5 | |||

${T}_{\left(\lambda \mu \nu \right)\hspace{0.17em}}$ | 3PA | 10 | 0 | 1 | 0 | 1 | 2 | 2 | |||

n = 4 | |||||||||||

${T}_{\lambda \mu \left(\nu \pi \right)}$ | 4WM | 54 | 2 | 3 | 4 | 2 | 1 | 34 | 23 | ||

${T}_{\left(\lambda \mu \right)\left(\nu \pi \right)}$ | OKE | 36 | 2 | 1 | 3 | 1 | 1 | 16 | 12 | ||

${T}_{\lambda \left(\mu \nu \pi \right)}$ | THS | 30 | 1 | 1 | 2 | 1 | 1 | 8 | 7 | ||

${T}_{\left(\lambda \mu \nu \pi \right)}$ | 4PA | 15 | 1 | 0 | 1 | 0 | 1 | 3 | 3 | ||

n = 5 | |||||||||||

${T}_{\left(\lambda \mu \nu \pi \rho \right)}$ | 5PA | 21 | 0 | 1 | 0 | 1 | 0 | 1 | 3 | 3 | |

n = 6 | |||||||||||

${T}_{\left(\lambda \mu \nu \pi \right)\left(\rho \sigma \right)}$ | SWM | 90 | 2 | 1 | 4 | 2 | 3 | 1 | 1 | 36 | 25 |

${T}_{\left(\lambda \mu \nu \pi \rho \sigma \right)}$ | 6PA | 28 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 4 | 4 |

**Table 2.**Combinations of weight that arise in processes involving up to four photons, in all common molecular point groups (those with up to six-fold rotational symmetry, and also the linear groups).

${T}_{\lambda \mu}$ | 210 | 21 | 20 | 2 | 1 | 0 | ||||||||

${T}_{\left(\lambda \mu \right)}$ | 20 | 2 | 0 | |||||||||||

${T}_{\lambda \mu \nu}$ | 3210 | 321 | 320 | 32 | 31 | 30 | 20 | 3 | 2 | 1 | 0 | |||

${T}_{\lambda \left(\mu \nu \right)\hspace{0.17em}}$ | 321 | 32 | 3 | 2 | 1 | |||||||||

${T}_{\left(\lambda \mu \nu \right)\hspace{0.17em}}$ | 31 | 3 | 1 | |||||||||||

${T}_{\lambda \left(\mu \nu \pi \right)}$ | 43210 | 4321 | 4320 | 432 | 431 | 430 | 420 | 43 | 42 | 40 | 4 | 3 | 1 | 0 |

${T}_{\left(\lambda \mu \nu \pi \right)}$ | 420 | 42 | 4 | 0 |

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Andrews, D.L.
Symmetries, Conserved Properties, Tensor Representations, and Irreducible Forms in Molecular Quantum Electrodynamics. *Symmetry* **2018**, *10*, 298.
https://doi.org/10.3390/sym10070298

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Symmetries, Conserved Properties, Tensor Representations, and Irreducible Forms in Molecular Quantum Electrodynamics. *Symmetry*. 2018; 10(7):298.
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2018. "Symmetries, Conserved Properties, Tensor Representations, and Irreducible Forms in Molecular Quantum Electrodynamics" *Symmetry* 10, no. 7: 298.
https://doi.org/10.3390/sym10070298