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Molecular Tensor Analysis of Third-Harmonic Scattering in Liquids Jack Stephen Ford, and David L. Andrews J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b10333 • Publication Date (Web): 06 Dec 2017 Downloaded from http://pubs.acs.org on December 20, 2017

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The Journal of Physical Chemistry

Molecular Tensor Analysis of Third-Harmonic Scattering in Liquids J. S. Ford and D. L. Andrews* School of Chemistry, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, United Kingdom

* [email protected]; +44 1603 592014

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Third-harmonic scattering is a nonlinear optical process that involves the molecular secondhyperpolarizability, γ. This work presents a rigorous quantum electrodynamical analysis of the scattering process, involving a partially index-symmetric construction of the fourth-rank γ tensor – dispensing with the Kleinman symmetry condition. To account for stochastic molecular rotation in fluids, methods of isotropic averaging must be employed to relate the molecular properties to accessible experimental quantities such as depolarization ratio. A complete eighth-rank tensor rotational average yields results for observable third harmonic scattering rates, cast as a function of the natural-invariant γ components, and the polarization geometry of the experiment. Decomposing the tensor γ into irreducible weights allows specific predictions to be made for each molecular point group, allowing greater discrimination between the results for different molecular symmetries.


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1. Introduction Rayleigh scattering is a commonplace optical phenomenon, consisting of the coupled annihilation and emission of a single photon by individual molecules. Its ubiquity is due to a linear dependence on input light intensity, and the fact that the E12 optical response (the polarizability, α) has an even parity, and hence a finite value for all materials. With high-intensity input light, some molecules can exhibit higherorder multiphoton scattering processes with the same incoherent character – distinct from the more familiar parametric (coherent, forward-emission) processes such as second and third harmonic generation. First, there is hyper-Rayleigh scattering, involving the annihilation of two photons and emission of a second-harmonic photon; this process is characterized by a quadratic dependence on intensity, and the odd parity of the E13 response (the hyperpolarizability, β).1–4 The next process in this sequence is third-harmonic scattering (THS), consisting of three annihilations and the incoherent emission of a third-harmonic photon; the THS rate has cubic dependence on intensity, and the E14 response (the second-hyperpolarizability, γ) has even parity, allowing any molecule to undergo this process.5,6 Improvements in optical technologies have made THS a practical technique for probing detailed molecular response properties. In two recent developments, a sensitive and efficient method of measuring the second-hyperpolarizability of solvents and solutes has been developed,7 and polarizationresolved experiments have allowed an interpretation in terms of irreducible components of the tensor.8 This article presents a quantum electrodynamical (QED) analysis of the THS process, describing the E14 interaction with fourth-order perturbation theory. A complete expression for the secondhyperpolarizability tensor is derived explicitly, without applying any general approximations regarding symmetry properties. In order to focus on scattering by molecular liquids, it is necessary to account for stochastic rotation. This signifies that the observed form of optical response is determined by an isotropic average of the molecular tensor – in low-viscosity liquids, individual molecules have no significant degree of persistent orientational or positional correlation. Several methods are available for 3 ACS Paragon Plus Environment


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calculating the appropriate, rotationally-averaged forms, and this paper outlines a complete eighth-rank tensor rotational average for THS, with results involving the natural-invariant components. Further development of the results into irreducible form provides the basis for a full discussion of tensor index symmetry and its consequences. In addition to general expressions for THS rate as a function of input light intensity and frequency, we provide geometry-specific predictions for the depolarization ratio and the reversal ratio. Our results, which include previously discounted components of the secondhyperpolarizability tensor, prove to offer a greater degree of discrimination between different molecular symmetries.

2. Molecular QED of THS Third-harmonic scattering is a four-interaction process – each molecule annihilates 3 photons of frequency ω and creates one photon of frequency 3ω. In a non-relativistic scheme, these four permutable matter-radiation interaction events unambiguously divide the history of the molecule-plusradiation system into five periods: the initial condition, three intermediate times, and the final outcome. i → φ1 → φ2 → φ3 → f

With q photons of frequency ω present initially, and the molecule’s electronic ground state labelled g, the initial and final states are i = g ; q;0 f = g ; ( q − 3 ) ;1

(1)

In the Dirac bras and kets of this section, the states of the system are labeled g ; m ; n = g m n . The first clause represents the state of the molecule, with the subsystem ket g designating the electronic state g; the second clause represents the state of the radiation mode distinguished by frequency ω, with the subsystem ket m indicating an occupation number of m photons; the third clause represents the 4 ACS Paragon Plus Environment

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state of the radiation mode distinguished by frequency 3ω, with the subsystem ket n indicating an occupation number of n photons. Without making any assumptions about the molecule’s structure, its electronic states in the intermediate times can be assigned the generic labels r,s,t. The radiation states in the intermediate times are determined by the time-ordering of the three ω photon-annihilation events and the 3ω photon-creation event. In general, a four-event process has 4! = 24 possible time-orderings, but here the three annihilated photons are indistinguishable so only four are distinct – these are illustrated in Figure 1 using non-relativistic Feynman diagrams. Perturbation theory gives the following expression for the quantum amplitude of such a four-event process.9 3 f Hˆ int φ3 φ3 Hˆ int φ2 φ2 Hˆ int φ1 φ1 Hˆ int i  Hˆ int  ˆ = M fi = f  ∑  Hint i φ1 ,φ2 ,φ3 ( Ei − Eφ3 ) ( Ei − Eφ2 )( Ei − Eφ1 )  Ei − H0 

(2)

The sum over all possible intermediate system states ϕ can be separated into a sum over the possibilities for intermediate molecule states (the possible identities of r,s,t) and a sum over the possibilities for intermediate radiation states (given by the four time-orderings): (2) (3) (4) M fi = ∑( M (1) fi + M fi + M fi + M fi ) r ,s,t

(3)

The complete kets i , φ1 , φ2 , φ3 and f for each time-ordering are given explicitly in Figure 1. The explicit form of the quantum amplitude terms for each time-ordering are as follows:



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Figure 1: Non-relativistic Feynman diagrams illustrating the THS process. The horizontal axis represents space; the vertical, time proceeding upwards; the kets to the right of each figure show how the interaction events divide history into five periods, and describe the system’s state in each period.

ˆ ˆ M (1) fi = g ; ( q − 3) ;1 H int t ; ( q − 3) ;0 t ; ( q − 3) ;0 H int s; ( q − 2 ) ;0 × s; ( q − 2 ) ;0 Hˆ int r; ( q − 1) ;0 r; ( q − 1) ;0 Hˆ int g ; q;0 × ( Egt + 3hω )

−1

(E

gs

+ 2hω )

−1

(E

gr

+ hω )

−1


(4)

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ˆ ˆ M (2) fi = g ; ( q − 3) ;1 H int t ; ( q − 2 ) ;1 t ; ( q − 2 ) ;1 H int s; ( q − 2 ) ;0 × s; ( q − 2 ) ;0 Hˆ int r; ( q − 1) ;0 r; ( q − 1) ;0 Hˆ int g ; q;0 × ( Egt − hω )

−1

(E

gs

+ 2hω )

−1

(E

+ hω )

gr

−1

(5)

ˆ ˆ M (3) fi = g ; ( q − 3) ;1 H int t ; ( q − 2 ) ;1 t ; ( q − 2 ) ;1 H int s; ( q − 1) ;1 × s; ( q − 1) ;1 Hˆ int r; ( q − 1) ;0 r; ( q − 1) ;0 Hˆ int g ; q;0 × ( Egt − hω )

−1

(E

gs

− 2hω )

−1

(E

+ hω )

gr

−1

(6)

ˆ ˆ M (4) fi = g; ( q − 3) ;1 Hint t ; ( q − 2 ) ;1 t ; ( q − 2 ) ;1 Hint s; ( q − 1) ;1 × s; ( q −1) ;1 Hˆ int r; q;1 r; q;1 Hˆ int g; q;0 ×( Egt − hω )

−1

(E

gs

− 2hω )

−1

(E

− 3hω )

gr

−1

(7)

The difference in energy between two molecular electronic states is written as, for example E gr ≡ E g − E r . The interaction-Hamiltonian operator Hˆ int describes the perturbation of the system by the

coupling of a certain quantized radiation mode with the molecule’s electromagnetic response. In the usual electric dipole (E1) approximation, only the electric dipole response need be included. For an interaction involving the radiation mode of frequency ω,10 1

Hˆ int

 hω  2 † = i  N (ω )  aˆ(ω ) e − aˆ(ω )e  ⋅ µˆ 2 V ε  0 

(8)

which is a general form accommodating both photon creation and annihilation terms; a directly similar expression applies to the harmonic (3ω) mode. Therefore, the Dirac brackets in equations (4-7) evaluate according to the quantum algebra of the appropriate photon creation and annihilation operators, aˆ (†3ω ) and aˆ(ω ) respectively. For the 3ω photon-creation event, 1

r ; q;1 Hˆ int

 3hω  2 rg g ; q; 0 = + i   N ( 3ω ) e′ ⋅ µ 2 V ε  0 


(9)


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and for each of the ω photon-annihilation events, 1

r; ( q − 1) ;0 Hˆ int

 q hω  2 rg g ; q;0 = −i   N (ω ) e ⋅ µ 2 V ε  0 

(10)

Here, V is the volume on average containing q photons of frequency ω; e is the polarization vector of the input ω mode; e′ is the polarization vector of the scattered 3ω mode; N is a local field correction factor describing the effects of the solvent; and µ rg is the molecule’s transition dipole moment. Fully evaluating equation (3), the quantum amplitude of the THS process is expressible as; 2

 hω  3 M fi = − 3q ( q −1)( q − 2)   N(3ω ) N(ω) ei′ e j ek el γ i( jkl )  2ε 0V 

(11)

where the subscripts i, j, k, l register Cartesian components. Note the Einstein convention of summation over repeated indices, such as u ⋅ v ≡ u j v j . The tensor γ i ( jkl ) represents the E14 molecular response, the molecule’s second-hyperpolarizability:

γ i( jkl ) =

1  ∑ 6 r , s ,t  

µigt

( Egt + 3hω )( Egs + 2hω )( Egr + hω ) + +

+

(E (E (E

gt

gt

rg j

µ ksr µlts + 5 { j ↔ k ↔ l} )

µits

(µ

rg j

µksr µlgt + 5 { j ↔ k ↔ l} )

µisr

(µ

rg j

µ kts µlgt + 5 { j ↔ k ↔ l})

− hω )( Egs + 2hω )( Egr + hω ) − hω )( Egs − 2hω )( Egr + hω )

µirg

gt

(µ

− hω )( Egs − 2hω )( Egr − 3hω )

(µ

sr j



µkts µlgt + 5 { j ↔ k ↔ l})  

(12)

All variables appearing in this γ i ( jkl ) definition are interpreted as real, so there is no imaginary part to the second-hyperpolarizability tensor as defined here. In particular, all of the energy differences are real, and the E1 transition dipole moments can always be constructed in a real form. But in general – in order to reflect the phenomenological form of response observed on any approach to resonance – imaginary 8 ACS Paragon Plus Environment

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damping coefficients may be included in the energy-difference denominators, rendering the tensor complex. For this reason, we are careful to distinguish γ i ( jkl ) from its complex conjugate, γ i( jkl ) . The four terms of γ i ( jkl ) correspond to the four terms of equation (3), and the transition dipole moments represent the molecular parts of the Dirac brackets in equations (4-7). The three transition moments derived from equation (10) each contract with the same polarization vector e, so the indices j,k,l that connect them to the three identical e vectors in equation (11) are degenerate. In each term, the triple contraction has thus been written as an average of 6 index-permutations, e.g. the first line of equation (12) contains:

 µ rgj µksr µlts + 5{ j ↔ k ↔ l}  rg sr ts rg sr ts   µ µ µ ⋅ ⋅ ⋅ = = e µ e µ e µ e e e e e e ( )( )( ) j k l  j k l  j k l   6  

(13)

Nonetheless the transition dipole moment from equation (9), assigned the index i, is necessarily distinct as it contracts with the scattered mode’s vector e′ . The tensor γ i( jkl ) as defined here is therefore indexsymmetric in j,k,l, but not in i. The brackets within the subscripts explicitly denote the permutational symmetry of three indices. This partial index-symmetry is a departure from the Kleinman condition of full permutational symmetry in all of the tensor’s indices – see section 8 for further discussion.

3. THS rate equation In general, the quantum amplitude Mfi for a process f←i is related to the observable rate of occurrence Γ by the Fermi rule,

Γ=

2 2π ρ f M fi = ∫ Γ′ d Ω h

(14)



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where the final states f of THS involve a photon of frequency 3ω propagating in some unspecified direction within an element of solid angle dΩ; Γ′ is the rate of directional emission per unit solid angle. The relevant density of states is:11

ρf =

9  ω 2V  8π 3  hc 3

 dΩ 

(15)

Separating the scalar and tensor parts of equation (11) gives the THS rate in tensor form:

Γ′ = Q ei′ e j ek el em′ en eoep γ i( jkl )γ m( nop)

(16)

The scalar Q subsumes ρ f , along with all scalar factors of Mfi. To more directly link to physically measurable input parameters, Q is best expressed in terms of the irradiance of the ω (absorbed) mode,

I = hcω

q ( 3) −2 , and a third-order coherence factor, g = ( q −1)( q − 2) q where the right-hand side is to be V

interpreted as an expectation value;

Q=

27 (3) 3 2 6  ω3  g I N(3ω) N(ω)  6 4  64π 2  hc ε0 

(17)

The g(3) factor can modify the cubic dependence on irradiance significantly, if the beam is not perfectly coherent. The N factors are best expressed in terms of the solvent’s refractive index at the relevant frequency, n(ω ) :12 −1

N (ω )

n(2ω ) + 2  2 ∂ n(ω )  2 =  n(ω ) + n(ω )ω  3  ∂ ω 

(18)

The rate may alternatively be expressed as radiant intensity of scattered radiation, I Ω′ = 3hω Γ′ .


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4. Rotational averaging To further develop the theory for application to liquids it is clearly necessary to refer the molecular response tensor to molecule-fixed coordinates, rather than the laboratory-fixed coordinates implied by the use of Roman-alphabet indices. The observed form of optical response is then determined by an isotropic average. There are various methods and levels of approximation that may be employed, such as have been reported by Wagnière13 and more recently by Friese et al.14 Here we prefer to adopt a method that can deliver results without applying any symmetry assumptions; our method of tensor rotational average is outlined in Appendix A.15,16 The THS rate equation (16) now becomes: (r ) (s) ei′ e j ek el em′ en eoep  mrs(8)  fιχκλµνοπ γι( χκλ )γ µ(νοπ )  Γ′ = Q  fijklmnop r s

(19)

Here, chevron brackets indicate an integration over the three Euler angles relating the two Cartesian coordinate systems (radiation- and molecule-fixed). The indices r and s each run through the integers (8)

1→105, so there are 11025 rate terms. The factor mrs consists of real number coefficients. Evaluating the scalars in square brackets, the 105 r values naturally form eight sets R that give common results for the field tensor; the 105 s values naturally form eight sets S that give common results for the molecular response tensor. The origin of this degeneracy is explained in Appendix A. (r ) r ∈ R ⇒  fijklmnop ei′ e j ek el em′ en eo ep  =  fe8  r R (s) s ∈ S ⇒  fιχκλµνοπ γ ι ( χκλ )γ µ (νοπ )  = [ f γγ ]S  s

(20)

The double-tensor symmetry causes the sets R and the sets S to have exactly the same numbered elements in r and in s respectively. Therefore it is convenient to give each set S the same alphabetic label as the corresponding set R: the eight labels A→H are assigned arbitrarily. The eight scalars [ f γγ

]S

the “natural invariants” of the molecular response tensor γι ( χκλ )γ µ(νοπ ) . This decomposition into eight linearly independent parts is irreducible, but not unique. 11 ACS Paragon Plus Environment

are


 fe8  R

[ f γγ ]S

( e⋅ e)( e⋅e′)( e⋅ e′)( e⋅ e) ( e⋅e)( e⋅e′)( e⋅e′)( e⋅e)

γ λ( ρλσ )γ σ ( ρνν )

( e⋅ e ) ( e⋅ e′)( e ⋅e′) 2 ( e⋅ e ) ( e⋅e′)( e ⋅ e′) 3 ( e⋅ e ) ( e′⋅ e′)

γ λ( λρσ )γ µ ( µρσ )

( e⋅e)( e ⋅ e)( e⋅ e′)( e′⋅ e) ( e⋅e)( e ⋅ e)( e⋅e′)( e ⋅ e′) ( e⋅e)( e ⋅ e)( e⋅ e)( e′⋅ e′)

γ λ( λρρ )γ µ ( µσσ )

R or S A B C D E F G H

2

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γ σ ( ρνν )γ λ( ρλσ ) γ µ ( λρσ )γ λ( µρσ ) γ µ ( λρσ )γ µ( λρσ ) γ µ ( λρρ )γ λ( µσσ ) γ µ ( λρρ )γ µ( λσσ )

Table 1: The scalar natural invariants of the radiation tensor and molecular tensor parts of equation (16).

(8)

(8)

The 105×105 matrix with components mrs reduces to an 8×8 matrix with components mRS by taking the total of the elements in each pair of sets.

(8) mRS =

∑

mrs(8)

r∈ R , s∈S

(21)

Thus the rotationally-averaged rate equation reduces to 64 terms: A →H

(8) Γ′ = Q ∑  fe8  mRS [ f γγ ]S R

(22)

R,S

The explicit matrix form of this rate equation is reported in Appendix A.

5. Tensor weights formulation

In order to address the dependence of the rate results on the detailed symmetry of the molecule, it is expedient to recast the molecular response tensor in terms of irreducible Cartesian tensor contributions.


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For a fourth-rank tensor, the irreducible parts take the form of natural tensors of ranks 0→4, embedded in the fourth-rank tensor space. The contributions that have rank j are called the “weight j” parts of the tensor. The first part of Appendix B outlines the method for decomposing a fourth-rank tensor. Importantly, the index-symmetry of the tensor naturally determines the number of irreducible parts of each weight that will appear.17 According to this scheme, the second-hyperpolarizability tensor decomposes as follows: (1) (2α ) (2 β ) (3) (4) γ i( jkl ) = γ i(0) ( jkl ) + γ i( jkl ) + γ i( jkl ) + γ i( jkl ) + γ i( jkl ) + γ i( jkl )

(23)

Each part is denoted by a superscript indicating its weight: γ i(( jjkl) ) is the tensor part of weight j. As shown in Appendix B, the partial index-symmetry of γ i ( jkl ) delivers one part each of weights 0, 1, 3 and 4, but two parts of weight 2 – the latter are to be distinguished by the labels 2α and 2β. In a completely general formulation, γ i ( jkl ) has 30 distinct Cartesian components. Since each part of weight j carries (2j + 1) independent elements, the above decomposition into six parts faithfully renders the total as 1 + 3 + 5 + 5 + 7 + 9 = 30. Each irreducible part of weight j has the transformation properties, under the symmetry operations of the full rotation group SO(3), of a fully index-symmetric, traceless tensor of rank j. Thus γ i((0jkl) ) transforms as a scalar; γ i(1) transforms as a vector, etc. The brackets in the ( jkl ) subscripts of these tensors refer to the index-symmetry of the γ i ( jkl ) tensor that they are parts of (as distinct from intrinsic, property-based tensor symmetry). To continue; in the rigorous weight formulation, the inner product of the second-hyperpolarizability with its complex conjugate, γ i ( jkl )γ i ( jkl ) , becomes an object of central interest. Of the 36 general terms that may arise from a product of two six-part tensors, only those of commensurate weight are non-vanishing, and so this scalar product in fact has just eight pure-weight scalar components.

γγ = γγ (0) + γγ (1) + γγ (2αα ) + γγ (2αβ ) + γγ (2βα ) + γγ (2ββ ) + γγ (3) + γγ (4)


(24)


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These scalars are denoted by the following shorthand:

γγ ≡ γ i ( jkl )γ i ( jkl ) γγ ( j) ≡ γ i((j)jkl ) γ i(( j)jkl ) γγ

(2αβ )

≡γ

( 2α ) i ( jkl )

γ

(25)

(2β ) i ( jkl )

With the γ i ( jkl ) definition (12) in use here, the scalar γγ and each of its eight pure-weight parts can be taken to be real and positive. The eight pure-weight components of γγ may now be expressed as a linear combination of the eight natural invariants [ f γγ

]S , as outlined in Appendix B. This linear combination transforms the

rotationally-averaged THS rate equation (22) into a function of the pure-weight components:  Γ′ = Q  

2 5 2 1 1  2   fe8   − γγ (2αα ) − γγ (2αβ ) − γγ (2 βα ) − γγ (2 ββ ) + γγ (3) − γγ (4)  A 35 14 35 14 42  35  5 2 2 1 1  2  +  fe8   − γγ (2αα ) − γγ (2αβ ) − γγ (2 βα ) − γγ (2 ββ ) + γγ (3) − γγ (4)  B 35 14 35 35 14 42   4 1 1 1 1 1   γγ (2αα ) + γγ (2αβ ) + γγ (2 βα ) + γγ (2 ββ ) − γγ (3) − γγ (4)  +  fe8   C 175 7 7 7 14 42   1  3  +  fe8   − γγ (3) + γγ (4)  D 12  28  1  3  +  fe8   γγ (3) + γγ (4)  E 28 36  

(26)

3 3 3 3 1 1  γγ (2αα ) + γγ (2αβ ) + γγ (2 βα ) − γγ (2 ββ ) + γγ (4)  +  fe8   γγ (0) − F 5 175 70 70 175 105   1 1 1 2 1 1  1  +  fe8   − γγ (1) + γγ (2αα ) + γγ (2αβ ) + γγ (2 βα ) + γγ (2 ββ ) + γγ (3) − γγ (4)  G 14 14 14 175 140 84  10  1 1 1 2 11 (3) 1 (4)  1 γγ (2 ββ ) − γγ − γγ  +  fe8   γγ (1) + γγ (2αα ) + γγ (2αβ ) + γγ (2 βα ) + H 10 14 14 14 175 140 84  

  

8 Each radiation scalar  fe  R engages with less than the full set of eight pure-weight components, so that

the number of distinct rate terms has been reduced from 64 to 42. This indicates that the weights formulation is the more natural representation of the rate calculation.


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6. Results: Application to specific experimental geometries

Equation (26) represents a complete, and perfectly general, expression for the rate of THS produced by a molecular liquid of arbitrary composition, with arbitrary directions and polarizations of input and scattered light. From it, we can now derive the explicit, much simpler, expressions for application to four typical experimental setups. Forward emission is usually swamped by the coherent third harmonic generation process, so these setups generally involve either right-angled or ‘backward’ beam geometries: •

“Parallel”: Linearly-polarized light is input; scattered light of parallel polarization is detected from an orthogonal position.

•

“Perpendicular”: Linearly-polarized light is input; scattered light of perpendicular polarization is detected from an orthogonal position.

•

“Preserved”: Circularly-polarized light is input; scattered light of the same polarization is detected in the reverse direction.

•

“Flipped”: Circularly-polarized light is input; scattered light of opposite circular polarization is detected in the reverse direction.

We note that scattered third-harmonic generation can be numerically removed (for example by exploiting the different angular dependences of THS and linear scattering of the third harmonic); we also observe that the coherent process of harmonic generation is forbidden for a circularly polarized input mode.18,19 The choice of experimental setup specifies the orientation of the polarization vectors of the input and output radiation modes. These vectors are taken to each have a modulus of unity. Accordingly, each is expressible as a combination of standard-basis unit vectors – see Table 2. Unit vector kˆ represents the propagation of the input ω mode; e is the polarization vector of the input ω mode; kˆ ′ is the propagation vector of the detected 3ω mode; e ′ is the polarization vector of the detected 3ω mode. 15 ACS Paragon Plus Environment


Parallel Perpendicular kˆ

zˆ

zˆ

e

xˆ

xˆ

kˆ ′

yˆ

yˆ

e′

xˆ

zˆ

Page 16 of 37

Preserved

Flipped

zˆ 1 ( xˆ + i yˆ ) 2 −zˆ 1 ( xˆ − i yˆ ) 2

zˆ 1 ( xˆ + i yˆ ) 2 −zˆ 1 ( xˆ + i yˆ ) 2

Table 2: Orientation of the radiation mode vectors in the four setups, expressed as combinations of the

standard-basis Cartesian unit vectors.

R

 fe8  R

Parallel value

Perpendicular value

Preserved value

Flipped value

A

( e⋅ e)( e⋅e′)( e⋅ e′)( e⋅ e)

1

0

0

0

B

( e⋅e)( e⋅e′)( e⋅e′)( e⋅e)

1

0

0

0

C

( e⋅ e ) ( e⋅ e′)( e ⋅e′)

1

0

0

1

D

( e⋅ e ) ( e⋅e′)( e ⋅ e′)

1

0

1

0

E

( e⋅ e ) ( e′⋅ e′)

1

1

1

1

F

( e⋅e)( e⋅ e)( e⋅ e′)( e′⋅ e)

1

0

0

0

G

( e⋅e)( e⋅ e)( e⋅e′)( e⋅ e′)

1

0

0

0

H

( e⋅e)( e⋅ e)( e⋅ e)( e′⋅ e′)

1

1

0

0

2

2

3

Table 3: Radiation natural invariant values, derived from Table 2.


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8 Each scalar  fe  consists of four dot products of polarization vectors. Within each experiment R

geometry, these scalars take specific values, listed in Table 3. Evaluating equation (26) with these coefficient values delivers THS rates for the four experimental setups:

6 (2αα ) 3 (2αβ ) 3 (2βα ) 6 (2ββ ) 8 1  = Q  γγ (0) + γγ − γγ − γγ + γγ + γγ (4)  175 35 35 175 315 5 

Γ′

Parallel

Γ′

Perpendicular

Γ′

Preserved

Γ′

Flipped

(27)

1 1 1 2 (2ββ ) 1 (3) 1 (4)  1 = Q  γγ (1) + γγ (2αα ) + γγ (2αβ ) + γγ (2βα ) + γγ + γγ + γγ  (28) 14 14 14 175 35 63 10 

1  = Q  γγ (4)  9 

(29)

1 1 1 1 1  4  γγ (2αα ) + γγ (2αβ ) + γγ (2βα ) + γγ (2ββ ) + γγ (3) + γγ (4)  = Q 7 7 7 28 252 175 

(30)

The result in which there is only a single contributing weight, as arises for the “preserved” geometry, deserves special comment. In this setup, the radiation-molecule interactions of the THS process involve the molecule taking in three units of optical angular momentum ℏ and outputting -1 unit (both the input ω mode and the output 3ω mode are left-polarized, and emission in the reverse direction has the same angular momentum effect as absorption), so that exactly four units are exchanged. The cylindrical symmetry of the system in this setup offers a unique axis of quantization (the experiment’s z axis), so strict conservation of angular momentum applies: the molecule must therefore undergo a four-unit transition in its rotational state, which is provided by the weight 4 component of the molecular response tensor γ i( jkl ) . In contrast, the “flipped” geometry involves interactions that can be interpreted as the molecule taking in three units of angular momentum and outputting +1 unit, thus engaging all weights in the range 3−1 → 3+1, by the usual rules of angular momentum coupling. Predictions for the exact THS depolarization ratio (DR) and reversal ratio (RR) are found by taking the ratios of the THS rate expressions in the relevant setups: 17 ACS Paragon Plus Environment


DR =

RR =

Γ′

Perpendicular

Γ′

Γ′ Γ′

=

Parallel

Flipped Preserved

=

315γγ (1) + 225 γγ (2αα ) + 225 γγ (2αβ ) + 225γγ (2 βα ) + 36 γγ (2 ββ ) + 90 γγ (3) + 50 γγ (4) 630 γγ (0) + 108 γγ (2αα ) − 270 γγ (2αβ ) − 270 γγ (2 βα ) + 108γγ (2 ββ ) + 80 γγ (4)

144 γγ (2αα ) + 900 γγ (2αβ ) + 900 γγ (2 βα ) + 900 γγ (2 ββ ) + 225 γγ (3) + 25 γγ (4) 700 γγ (4)

Page 18 of 37

(31)

(32)

These ratios are experimentally accessible by comparing the THS intensities measured with the two specified detector geometries, keeping all other factors constant. The scalar Q cancels out, so that these results are independent of whether the detected intensity of THS is formulated in terms of an absolute rate Γ, or directional rate per unit solid angle Γ′, or radiant intensity IΩ′ , etc. We can now develop the above results for application to molecules of specific symmetry classes, enabling predictions to be made for each molecular point group.

7. Results: Consequences of molecular symmetry

Equations (31) and (32) are the general equations describing a THS process where the secondhyperpolarizability contains all weights 0→4, such that the key scalar γγ has all eight components given by equation (24). The six weight parts from which these are constructed, described by equation (23), are intrinsic molecular properties, with values determined by the state-dependent energies and transition dipole moments appearing in the γ i ( jkl ) definition (12). For example, near-resonance enhancement of THS can be inferred from the inflation of γ i ( jkl ) when the energy differences in the denominators of (12) approach zero – nonetheless, no explicit simplification of the symmetry structure arises under such conditions. It can be shown that molecules of high symmetry are forbidden from having γ i ( jkl ) parts of certain weights, according to their point group – see Table 4 for a summary (weight data taken from the 4+ column of Table A6.2 in Ref.20). For molecules with such symmetry restrictions, the γγ components of 18 ACS Paragon Plus Environment

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forbidden weights must be set to zero, equivalent to deleting certain terms. What follows are the reduced results for DR and RR in such cases.

Point group C1 Ci Cs C2→C6 S4→S10 C2h→C6h C2v C3v C4v→C6v D2 D3 D4→D6 D2h D3h→D6h D2d D3d D4d→D6d C∞v D∞h T Th Td O Oh I Ih

Example molecule NHFCl meso (HOCH)2(COOH)2 CHF2Cl PPh3 CPh4 planar B(OH)3 H2 O NH3 F4VO C10H16 twistane [Co(N2C2H8)3]3+ Ni(CH2)4 C2H4 C6H6 benzene H2CCCH2 staggered [O3SSO3]2− staggered Fe(C5H5)2 HCN CO2 N4(C2H4)6 [Mg(H2O)6]2+ CCl4 N8(C2H4)12 SF6 C140 C60 buckminsterfullerene

γ i ( jkl ) weights 01234 01234 01234 01234 01234 01234 0 234 0 234 0 2 4 0 234 0 234 0 2 4 0 234 0 2 4 0 2 4 0 234 0 2 4 0 2 4 0 2 4 0 34 0 34 0 4 0 4 0 4 0 . 0 .

Table 4: Weights allowed under the totally-symmetric irreducible representation of molecular point

groups in the γ i ( jkl ) tensor.

First, let us consider the case with the most severe constraint, in which the second-hyperpolarizability has only a weight 0 part. In this case Γ ′ = 0 (i.e. THS is forbidden) in the “perpendicular”, “preserved”, 19 ACS Paragon Plus Environment


Page 20 of 37

and “flipped” geometries. This means, for example, that there is no back-scattering of third-harmonic circularly-polarized light, and DR = 0. These results apply to the point group Ih, describing the fullerene molecule C60. With some only slightly less symmetric molecules, the two extreme weights of 0 and 4 contribute. This is true for the point group Oh, describing the molecule SF6. Then, the polarization ratios reduce to:  63  γγ (0)  8  DR0,4 =   (4)  +   5  γγ  5 

RR 0,4 =

−1

(33)

1 28

(34)

Notably, the reversal ratio takes its lowest possible value for molecules of this high symmetry; it is evident from the form of the general result (32) that all other cases produce a higher RR value. It can also be observed that the limit of DR = 5/8 can only result from weight 4 being the dominant part of the (4) second-hyperpolarizability, γ i( jkl ) ≈ γ i( jkl ) , which is unlikely because the presence of weight 4 is always

accompanied by weight 0. Thus it also emerges that 5/8 is the maximum attainable value for DR. The most common case for simple molecules is where all three even weights contribute. For convenience, we can simplify the results involving weight 2 by naively combining the four weight-2 components of γγ into a single term, γγ

(2)

≡ γγ

( 2 αα )

+ γγ

( 2α β )

+ γγ

( 2 βα )

+ γγ

( 2 ββ )

. (There is no rigorous

procedure for unambiguously combining the four weight-2 components into a single meaningful term, because any such set of linearly independent elements can always be recast as an alternative set through an appropriate transformation.) With this chosen definition of γγ (2) ,

DR 0,2,4 =

711 γγ ( 2 ) + 50 γγ ( 4 ) 630 γγ ( 0) − 324 γγ ( 2 ) + 80 γγ

R R 0,2,4 =

711  γγ ( 2 )  1  + 175  γγ ( 4 )  28

(35)

(4)

(36)


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The appearance of weight 3 without weight 2 is relatively unusual. An example is the point group T, describing for example the tetrahedral chain molecule N4(C2H4)6, or the conformation-averaged substituted neopentane C(CHFCl)4 if the four outer carbons all share the same absolute configuration. For such molecules,

DR 0,3,4 =

RR0,3,4 =

9 γγ (3) + 5 γγ ( 4) 63 γγ (0) + 8 γγ ( 4)

(37)

9  γγ (3)  1  + 28  γγ (4)  28

(38)

8. Discussion: Index symmetry

The rotationally-averaged rate results (27) and (28) partially concur with a recent treatment of the problem by Rodriguez:

1 4 8 Parallel Γ′ (Rodriguez) ∝ γγ (0) + γγ (2) + γγ (4) 5 35 315 Perpendicular

Γ′ (Rodriguez) ∝

(39)

3 (2) 1 (4) γγ + γγ 140 63

(40)

(Adapted from equation (5) in Ref.8) The contributions of weight 0 and weight 4 agree exactly, but there is difference in the weights 1→3. Rodriguez’s THS theory is based on an analysis by Kielich et al,6,21 which applies the Kleinman symmetry approximation to the second-hyperpolarizability. If we were to assume full index-symmetry in our construction of the γ tensor, then the decomposition into weights according to equation (23) would instead take the form of equation (53) in Appendix B, with no odd weights. This difference lies at the root of the divergence of our results from those of other authors. In a rich body of work on molecular nonlinear optics, Kielich et al. were typical of an earlier chemical physics community when they assumed the general validity of the Kleinman symmetry condition, 21 ACS Paragon Plus Environment


having commented in passing that “this approximation is justified if the frequencies of the incident and scattered waves do not fall within regions of electron dispersion and absorption.”6 It is, undoubtedly, an approximation that leads to much greater simplicity in many theoretical formulations. However, the inexactitude of this approach has long since been recognized.22,23 The severity of assuming full index symmetry can be assessed from a variety of perspectives. For example, in a commonly used two-level approximation (where the analytical form of the tensor assumes a simple dependence on the dipolar properties of the ground state and one excited state – which must be accessible by single-photon absorption), the tensor acquires full index symmetry only if there is no difference between the permanent electric dipoles of the two states, i.e. if no charge redistribution occurs in the associated excitation transition.24 The unlikeliness of fulfilling such a condition is strikingly apparent when it is recalled that the same two-level approximation for the second-harmonic (hyperpolarizability) tensor renders the entire response zero, unless there is such a dipole difference.25 In fact, a great deal of the literature on organic materials for nonlinear optics rests upon the premise that the sought response is largely driven by a ‘push-pull’ dynamic, associated with donor and acceptor chromophore groups supporting such a dipole shift.26–29 More generally, for any optical frequency-multiplication process, entailing n events of ω-photon annihilation and one nω-photon creation, the molecular response χ(n) is an (n+1)th-rank tensor analogous to the n = 3 case described by equation (12).30–32 Simpson comments on the generic sum-frequency case: “Irrespective of the presence or absence of Kleinman symmetry, all nonlinear optical polarizability tensors exhibit intrinsic permutation symmetry, in which the χ(n) tensor is unchanged by the simultaneous exchange of any two frequency terms and the indices corresponding to those frequencies. In the case of [n=2], this intrinsic permutation condition can be written in a simple form; χ ijk ( −ωsum ; ωa , ωb ) = χ jik (ωa ; −ωsum , ωb ) = χ kji (ωb ; ωa , −ωsum ) , etc. […] the Kleinman symmetry condition is only generally maintained in the limit of all [frequencies] approaching a value of zero, in which case the recovery of overall permutation symmetry within the χ(n) tensor becomes trivial.”23 In this view, the Kleinman 22 ACS Paragon Plus Environment

Page 22 of 37

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approximation is equivalent to assuming that the molecular response does not discriminate between any of the energy differences −nℏω, −(n−1)ℏω, ... +(n−1)ℏω, and +nℏω. Third-harmonic scattering (n = 3) should be more likely to exhibit a violation of this condition than second-harmonic scattering (n = 2). For this reason, one can be confident that the partially-symmetric expressions reported in this article will be appropriate and necessary for the full interpretation of future THS experiments.

9. Conclusion

We have shown that, for reasons of physical realism, the analysis of THS warrants accounting for all weights 0→4 in the molecular response tensor. A rigorous formulation which includes the complete tensor behavior of the E14 molecular response, dispensing with the Kleinman symmetry condition, leads to results that allow specific predictions to be made for each molecular point group. The experimental utilization of these results should provide for greater discrimination between the observed behavior associated with different molecular symmetries. This improvement becomes increasingly necessary as THS develops into a more widespread molecular characterization technique.

Appendix A. Eighth-rank rotational average

Consider a generic rate equation similar to (16), formed from the contraction of an eighth-rank radiation tensor G and an eighth-rank molecular tensor Λ. Roman-alphabet indices signify components in a spacefixed Cartesian coordinate frame (x, y, z). Γ′ = Q Gijklmnop Λijklmnop

(41)

A molecular tensor is naturally expressed with orientation-invariant components in a molecule-fixed Cartesian frame. The molecule-fixed components of eighth-rank tensor Λ are labelled with Greek-



Page 24 of 37

alphabet indices, Λιχκλµνοπ . These are expressible as components in the space-fixed frame using the cosines of corresponding index pairs, l. Λijklmnop = li ι l j χ lk κ llλ lmµ lnν loο l pπ Λιχκλµνοπ

(42)

An isotropic rotational average entails an integration over the three Euler angles relating the two coordinate systems. This operation is denoted by chevron brackets – where X is any function of molecular orientation, X ≡∫

2π

0

π

2π

0

0

∫ ∫

sin θ X dφ dθ dψ 8π 2

(43)

The space-fixed representation of G and the molecule-fixed representation of Λ are invariant to this operation, therefore the rotational average of rate yields: Γ ′ = Q Gijklmnop li ι l j χ lk κ ll λ lmµ lnν loο l pπ Λιχκλµνοπ

(44)

From Weyl’s theorem, it is possible to decompose such a double-tensor into a linear combination of isotropic tensor products in which the Roman and Greek indices do not mix.15 li ι l j χ lk κ ll λ lmµ lnν loο l pπ =

1→105

∑

(r ) (s) fijklmnop mrs(8) fιχκλµνοπ

(45)

r, s

Each f is an isotropic tensor consisting of four Kronecker deltas that connect pairs of indices. There are 105 possible pairings of eight indices, e.g. (1) fijklmnop = δijδklδmnδop (2) fijklmnop = δijδklδmoδ np

M (105) fijklmnop = δipδ joδknδlm

(46)

The m(8) tensor is a 105×105 square matrix of real numbers. The value of each component mrs(8) is (r ) (s) determined by the factorisation structure of the contraction f ijklmnop , for example: f ijklmnop


Page 25 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(1) (105) f ijklmnop f ijklmnop = (δ ijδ joδ opδ ip ) (δ klδ lmδ mnδ kn )

(47)

The fact that the eight Kronecker deltas of this product factorise into two index-cycles of four is (8) diagnostic of the values m1;105 may take. In general this procedure does not uniquely determine the value

of each mrs(8) due to over-completeness issues, but we have chosen values that exploit linear dependences within the set of tensors f to give the simplest form of m(8).16 These are listed in Table 5. (r ) (s) factorisation f ijklmnop f ijklmnop

(δ δ) (δ δ) (δ δ) (δ δ) (δ δ δ δ) (δ δ) (δ δ)

mrs(8) 19/630

-23/3780

(δ δ δ δ) (δ δ δ δ)

1/7560

(δ δ δ δ δ δ) (δ δ)

1/756

(δ δ δ δ δ δ δ δ)

0

Table 5: Elements of the matrix m(8), derived from the corresponding isotropic tensors f.

Therefore, the rotationally-averaged rate equation (44) expands as 11025 terms. (r ) (s) Γ′ = Q  fijklmnop Gijklmnop  mrs(8)  fιχκλµνοπ Λιχκλµνοπ  r s

(48)

In the case of equation (16) describing third-harmonic scattering, Gijklmnop = ei′ e j ek el em′ en eo e p and Λιχκλµνοπ = γ ι ( χκλ )γ µ (νοπ ) . These tensors have partial index-symmetry: they are invariant to any exchange

of indices within the sets (jkl), (nop), (χκλ), and (νοπ). Consequently, any Kronecker delta contractions that differ by engaging alternate members of these sets will be equal: e.g. διχ γ ι ( χκλ ) = δικ γ ι ( χκλ ) , and

δ χκ γι( χκλ ) = δ χλγι( χκλ) . From the f examples given in equation (46), it is evident that r = 1 and r = 2 yield the same result for the THS radiation scalar. (r ) r ∈{1, 2} ⇒  fijklmnop ei′ e j ek el em′ en eo ep  = ( e ⋅ e )( e ⋅ e )( e ⋅ e′)( e′ ⋅ e ) r


(49)


Page 26 of 37

In fact there are seven other r values which also yield this result, and these nine integers comprise the set 8 labelled F, hence this value for the radiation scalar is designated  fe  . The other sets A→H are

F

constructed similarly, with elements listed in Table 6. R or S

Elements r of R; elements s of S

A

{4,5,6,7,10,13,19,20,21,22,25,28,34,35,36,37,40,43}

B

{62,63,65,66,70,73,77,78,80,81,85,88,92,93,95,96,100,103}

C

{8,9,11,12,14,15,23,24,26,27,29,30,38,39,41,42,44,45}

D

{68,69,71,72,74,75,83,84,86,87,89,90,98,99,101,102,104,105}

E

{53,54,56,57,59,60}

F

{1,2,3,16,17,18,31,32,33}

G

{61,64,67,76,79,82,91,94,97}

H

{46,47,48,49,50,51,52,55,58}

Table 6: Integer values of r and s comprising the sets R and S described by equation (20).

(8) The reduced m(8) tensor, with components mRS defined by equation (21), is expressible as an 8×8

matrix.

m (8)

 180   72  −90  1  −90 = 1260  60   −18  −18   −72

72

−90 −90

60 −18

180

−90 −90

60

−90 180 −90

60

45

−75

180

−75

−75 −75

110

45

−18 −18

9

3

−18

9

−18

3

−72

63

63

−78

−18 −72   −18 −18 −72  9 63  −18  9 −18 63  3 3 −78   72 −9 −9  72 −9  −9  −9 −9 126 

(50)

Using this matrix form, the 64-term rate equation (22) is a row-square-column matrix multiplication.


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T

  fe8     A   [ f γγ ]A    fe8       B   [ f γγ ]B    fe8    [ f γγ ]    C  C     fe8   f γγ [ ]     D D  m(8) Γ′ = Q   . 8   fe    [ f γγ ]E    E   [ f γγ ]    8  F     fe  F  f γγ [ ]   G  8  fe     G   [ f γγ ]H   8   fe    H 

(51)

Appendix B. Irreducible tensors and a linear combination of the natural invariants ( j ,q ) A generic non-symmetric forth-rank tensor Π naturally decomposes into 19 irreducible parts Π ijkl ,

identified by weight j and by seniority q.17 For the full expressions for these tensors in terms of the 81 Cartesian components of Π, see equations (3.1)-(3.19) in Ref.17. (0,1) (0,2) Π ijkl = Π ijkl + Π ijkl + Π (0,3) ijkl

(1,1) (1,2) (1,3) (1,4) (1,6) + Π ijkl + Π ijkl + Π ijkl + Π ijkl + Π (1,5) ijkl + Π ijkl (2,2) (2,3) (2,4) (2,5) (2,6) + Π (2,1) ijkl + Π ijkl + Π ijkl + Π ijkl + Π ijkl + Π ijkl

+Π

(3,1) ijkl

+Π

(4,1) ijkl

+Π

(3,2) ijkl

+Π

(52)

(3,3) ijkl

If a tensor has any degree of index-symmetry, the irreducible parts are given by combining some (but in general not all) of these 19 general terms, according to Table III in Ref.17. For example, consider a fully index-symmetric fourth-rank tensor, Ξ. According to this procedure it has a weight 0 part, a weight 2 part, and a weight 4 part: (2) (4) Ξ ( ijkl ) = Ξ ((0) ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl )

(53)

These are each constructed from a subset of the 19 general terms.



Page 28 of 37

(0,1) (0,2) (0,3) Ξ (0) ( ijkl ) = Ξ ( ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl )

(54)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6) Ξ (2) ( ijkl ) = Ξ ( ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl ) + Ξ ( ijkl )

(55)

(4,1) Ξ (4) ( ijkl ) = Ξ ( ijkl )

(56)

Now consider a fourth-rank tensor T that is symmetric in only its first three indices, with Cartesian components T( ijk ) l . The same procedure yields six irreducible parts: one each of weights 0, 1, 3 and 4, and two of weight 2. (1) (2α ) (2 β ) (3) (4) T( ijk ) l = T((0) ijk ) l + T( ijk ) l + T( ijk ) l + T( ijk ) l + T( ijk ) l + T( ijk ) l

(57)

Note that each of the six parts is a fully index-symmetric tensor, of rank equal to its weight, embedded in the fourth-rank tensor space as T( ijk ) l . Explicitly, these are: (0,1) (0,2) (0,3) T((0) ijk ) l = T( ijk ) l + T( ijk ) l + T( ijk ) l

=

1 (δ ijδ kl + δ ik δ jl + δ ilδ jk ) T(σρρ )σ 15

(58)

(1,3) (1,5) (1,6) T((1) ijk ) l = T( ijk ) l + T( ijk ) l + T( ijk ) l

=

1 (δ jk T(iττ )l − δ jk T(lττ )i + δ ik T( jττ )l − δ ik T(lττ ) j + δ ijT( kττ )l − δ ijT(lττ ) k ) 10

(59)

α) (2,1) (2,2) (2,4) T((2 ijk ) l = T( ijk ) l + T( ijk ) l + T( ijk ) l

=

−1 (δ ijδ kl + δ ik δ jl + δ ilδ jk )T(σρρ )σ 21 1 − (δ klT( i ρρ ) j + δ kl T( j ρρ ) i + δ jl T( i ρρ ) k + δ jl T( k ρρ ) i + δ il T( j ρρ ) k + δ il T( k ρρ ) j ) 21 5 + (δ jk T( i ρρ ) l + δ jk T( l ρρ ) i + δ ik T( j ρρ ) l + δ ik T( l ρρ ) j + δ ijT( k ρρ ) l + δ ij T( l ρρ ) k ) 42


(60)

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β) (2,3) (2,5) (2,6) T((2 ijk ) l = T( ijk ) l + T( ijk ) l + T( ijk ) l

=

−1 (δ ijδ kl + δ ik δ jl + δ ilδ jk ) T(σρρ )σ 21 5 + ( δ kl T( ij ρ ) ρ + δ jl T( ik ρ ) ρ + δ il T( jk ρ ) ρ ) 21 2 − (δ jk T( il ρ ) ρ + δ ik T( jl ρ ) ρ + δ ijT( kl ρ ) ρ ) 21

(61)

(3,3) T((3) ijk ) l = T( ijk ) l

3 1 = T(ijk ) l − (T( jlk ) i + T(ilk ) j + T( ijl ) k ) 4 4 1 1 + (δ jk T( ilτ )τ + δ ik T( jlτ )τ + δ ijT( klτ )τ ) − (δ kl T( ijτ )τ + δ jlT( ikτ )τ + δ liT( jkτ )τ ) 6 6 1 + (δ kl T(iνν ) j + δ klT( jνν )i + δ jl T( iνν ) k + δ jlT( kνν ) i + δ il T( jνν ) k + δ il T( kνν ) j ) 12 1 11 + (δ jk T( lνν ) i + δ ik T(lνν ) j + δ ijT( lνν ) k ) − (δ jk T( iνν ) l + δ ik T( jνν ) l + δ ij T( kνν ) l ) 60 60

(62)

(4,1) T((4) ijk ) l = T( ijk ) l

=

1 (T(ijk )l + T( jkl )i + T(ikl ) j + T(ijl )k ) 4 1 − (δ jk T(iνν )l + δ jk T( lνν )i + δ ik T( jνν )l + δ ik T( lνν ) j + δ ijT( kνν ) l + δ ijT(lνν ) k 28 + δ il T( jνν ) k + δ il T( kνν ) j + δ jl T(iνν ) k + δ jl T( kνν )i + δ kl T(iνν ) j + δ kl T( jνν ) i ) 1 (δ jkT(ilτ )τ + δ ikT( jlτ )τ + δ ijT( klτ )τ + δ klT(ijτ )τ + δ jlT(ikτ )τ + δ ilT( jkτ )τ ) 14 1 + (δ ijδ kl + δ ik δ jl + δ il δ jk ) T(σρρ )σ 35

−

(63)

The inner product of the tensor T( ijk ) l with its complex conjugate has eight scalar terms, each arising from a contraction of two equal-weight parts. This is because contractions of non-commensurate weight (1) combinations are vanishing, e.g. T((0) ijk ) l T( ijk ) l = 0 .

(0) (1) (1) (2α ) (2α ) (2α ) (2 β ) T( ijk )l T(ijk )l = T((0) ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l

β ) (2α ) (2 β ) (2 β ) (3) (3) (4) (4) + T((2 ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l + T( ijk ) l T( ijk ) l

(64)

By substituting equations (58)-(63) into (64) and performing the contractions, the eight pure-weight terms of T( ijk ) l T( ijk ) l evaluate as follows:



1 (0) T((0) T(σρρ )σ T( λµµ ) λ ijk ) l T( ijk ) l = 5

(1) T((1) ijk ) l T( ijk ) l =

Page 30 of 37

(65)

3 3 T( ρττ )σ T( ρνν )σ − T( ρττ )σ T(σνν ) ρ 10 10

(66)

α ) (2α ) T((2 ijk ) l T( ijk ) l =

−5 15 15 T(σρρ )σ T( λµµ ) λ + T(σρρ ) λ T(σµµ ) λ + T(σρρ ) λ T( λµµ )σ 21 42 42

(67)

α ) (2 β ) T((2 ijk ) l T( ijk ) l =

2 6 T(σρρ )σ T( λνν ) λ − T(πρρ )τ T(πτλ ) λ 21 21

(68)

β ) (2α ) T((2 ijk ) l T( ijk ) l =

2 6 T(σρρ )σ T( λνν ) λ − T(πτρ ) ρ T(πνν )τ 21 21

(69)

β ) (2 β ) T((2 ijk ) l T( ijk ) l = −

5 15 T(σρρ )σ T( λµµ ) λ + T(πτρ ) ρ T(πτµ ) µ 21 21

3 3 1 1 (3) T((3) T( ρπτ ) λ T( ρπτ ) λ − T( ρπτ ) λ T( ρπλ )τ + T( ρττ )π T( ρπλ ) λ − T( ρπτ )τ T( ρπλ ) λ ijk ) l T( ijk ) l = 4 4 2 2 1 1 11 + T( ρπτ )τ T( ρλλ )π + T( ρττ )π T(πλλ ) ρ − T( ρττ )π T( ρλλ )π 2 20 20

(4) T((4) ijk ) l T( ijk ) l =

−3 3 3 3 T( ρπτ )τ T( ρλλ )π − T( ρττ )π T( ρπλ ) λ − T( ρπτ )τ T( ρπλ ) λ + T( ρπτ ) λ T( ρπλ )τ 14 14 14 4 1 3 3 3 + T( ρπτ ) λT( ρπτ ) λ + T( ρρτ )τ T(ππλ ) λ − T( ρττ )π T(πλλ ) ρ − T( ρττ )π T( ρλλ )π 4 35 28 28

(70)

(71)

(72)

These results are directly applicable to third-harmonic scattering since our abstract tensor T( ijk ) l is only a single index-permutation away from the second-hyperpolarisability: T(ijk )l ≡ γ l ( ijk )

(73)

Through this transformation, equation (57) becomes identifiable as (23), and equation (64) as (24). The scalars appearing in the right-hand sides of equations (65)-(72) are equivalent to the eight natural invariants [ f γγ ]S listed in Table 1. Converting each T( ijk ) l into a γ l (ijk ) turns the set of eight equations


Page 31 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(65)-(72) into a linear combination for γγ in terms of the natural invariants. The left-hand sides of the following equations use the shorthand of equation (25). γγ (0) =

γγ

(1)

=

1 [ f γγ ]F 5

−3 [ f γγ 10

γγ (2αα ) =

γγ (2αβ ) =

γγ (2 βα ) =

γγ (2 ββ ) =

γγ (3) =

]G +

3 [ f γγ 10

]H

(75)

−5 [ f γγ 21

]F + [ f γγ ]G + [ f γγ ]H

−2 [ f γγ 7

]B +

2 [ f γγ 21

]F

−2 [ f γγ 7

]A +

2 [ f γγ 21

]F

5 14

5 14

(76)

(77)

(78)

5 5 [ f γγ ]C − [ f γγ ]F 7 21

1 [ f γγ 2

γγ (4) = −

(74)

1 2

1 2

(79) 3 4

3 4

]A + [ f γγ ]B − [ f γγ ]C − [ f γγ ]D + [ f γγ ]E +

1 [ f γγ 20

]G −

3 3 3 3 1 3 [ f γγ ]A − [ f γγ ]B − [ f γγ ]C + [ f γγ ]D + [ f γγ ]E + [ f γγ 14 14 14 4 4 35

11 [ f γγ 20

]F −

]H

3 [ f γγ 28

(80)

]G −

3 [ f γγ ]H 28 (81)

As a check of these results, note that the sum of all eight simultaneous equations (74)-(81) gives the identity γγ = [ f γγ ]E . This is correct, as

γ i ( jkl )γ i ( jkl ) = γ µ( λρσ )γ µ( λρσ ) indicates nothing but a change in

index labels. This linear combination is expressible in matrix form:



 [ f γγ ]A   γγ (0)     (1)   [ f γγ ]B   γγ   [ f γγ ]   γγ (2αα )  C    (2αβ )  f γγ  [ ]D   γγ    (2 βα )  = W   [ f γγ ]E   γγ   [ f γγ ]   γγ (2 ββ )  F     (3)  [ f γγ ]G   γγ     γγ (4)     [ f γγ ]H 

Page 32 of 37

(82)

The elements of square matrix W are given directly by the coefficients in the eight linear equations (74)(81), 0 0 0  0  0 0 0  0  0 0 0 0  0 120 0 0 − 1  W= 0 0 420  −120 0  0 300 0  0  210 210 −210 −315   −90 −90 −90 315 

0 84 0 0 0 −126 0 −100 150 0 40 0 0 40 0 0 −100 0 315 0 21 105 36 −45

0   126  150   0  0   0  −231   −45 

(83)

The 64-term rate equation (22), appearing in matrix form as equation (51), may be rearranged to give the rate in “pure-weight” form. T

  fe8     A   γγ (0)    fe8       B  γγ (1)   8   fe    γγ (2αα )    C   (2αβ )    fe8      D  (8) −1  γγ  m W  (2 βα )  Γ′ = Q   fe8    γγ    E  (2 ββ )   γγ   8      fe  F  (3)  γγ   8   γγ (4)    fe  G     8   fe    H 


(84)

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The composite square matrix m (8) W −1 is derived from equations (50) and (83) via square matrix inversion and square-square matrix multiplication. These calculations were performed using the free online matrix calculator at matrix.reshish.com. −360 −360 −2250 −360 450 −150   0 0   −360 −2250 −360 −360 450 −150   0 0  0 0 144 900 900 900 −450 −150    0 0 0 0 −675 525  1  0 0 (8) −1 m W = 0 0 0 0 0 675 175  6300  0   270 −108 0 60  −108 270  1260 0  0 −630 450 450 450 72 45 −75    − − 0 630 450 450 450 72 495 75  

(85)

The presence of 22 zero elements shows the reduction in the number of independent terms in the rate calculation. Evaluating equation (84) as a row-square-column matrix multiplication yields the 42-term rate equation (26), with coefficients given directly by the elements of matrix (85).

References

(1)

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(2)

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(3)

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(5)

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(6)

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(7)

Van Steerteghem, N.; Clays, K.; Verbiest, T.; Van Cleuvenbergen, S. Third-Harmonic Scattering for Fast and Sensitive Screening of the Second Hyperpolarizability in Solution. Anal. Chem. 2017, 89, 2964–2971.

(8)

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(9)

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