Polarizabilities. Patrick Norman

Transcription

1 Polarizabilities Patrick Norman Division of Theoretical Chemistry and Biology School of Biotechnology KTH Royal Institute of Technology Stockholm, Sweden A perspective on nonresonant and resonant electronic response theory for time-dependent molecular properties P. Norman, Phys. Chem. Chem. Phys. 13, (2011)

2 Polarizabilities and light scattering A time-dependent electric dipole will be the source of radiation, much like a microscopic antennas. Let us consider a point electric dipole described by the electric dipole density p(r, t) = µe iωt δ(r) where the complex notation is used for mathematical convenience and it is the real part that corresponds to the physical dipole moment. The point electric dipole corresponds to a current density j(r, t) = p(r, t) t = iωµe iωt δ(r)

3 Given the current density, we determine the vector potential in the Lorentz gauge from A(r, t) = µ 0 j(r, t r ) 4π r r d 3 r ; t r = t r r /c to arrive at A(r, t) = iωµ 0 4πr The magnetic field then becomes B(r, t) = A(r, t) = ω2 1 4πε 0 c 3 r µe iω(t r/c) ( (r µ) 2 where we have used the vector operation [af (r)] = (r a) 1 r 1 + ic ωr f (r). r ) e iω(t r/c)

4 From the Lorentz gauge condition, we have t φ = c 2 A(r, t) = where we have used ω2 (r µ) 4πε 0 cr 2 [af (r)] = (r a) 1 r ( 1 + ic ωr f (r) r ) e iω(t r/c) By means of time integration, we find that the retarded scalar potential is φ(r, t) = iω ( 1 (r µ) 1 + ic ) e iω(t r/c) 4πε 0 c r 2 ωr

5 The electric field becomes E(r, t) = φ t A ω 2 ( ) (r µ)r = 4πε 0 c 2 µ r r 2 e iω(t r/c) + 1 ( 1 4πε 0 r 3 iω ) ( ) (r µ)r cr 2 3 r 2 µ In the far-field limit (r large), we get E(r, t) = B(r, t) = ω 2 ( 4πε 0 c 2 µ r ) (r µ)r r 2 e iω(t r/c), ω 2 4πε 0 c 3 (r µ) e iω(t r/c) r 2 e iω(t r/c)

6 Scattering intensity and polarizabilities The associated time-averaged Poynting vector (or intensity) for complex fields becomes S T = 1 2 ε 0c 2 Re (E B ) = = ω 4 µ 2 32π 2 ε 0 c 3 r r 3 sin2 θ where θ is the angle between the induced molecular dipole µ and the direction of observation r. This scattering intensity is the microscopic origin of the observed photons in both Rayleigh and Raman scattering processes. For molecules with nonzero β( 2ω; ω, ω), the point dipole will emit radiation at a frequency 2ω. The corresponding spectroscopy is known as hyper-rayleigh scattering (HRS) and was designed to measure β for isotropic samples.

7 Physically correct polarizabilities at resonant frequencies b ΔE = 0.5 a.u. a µ = 1.0 a.u. ab ψ(0) = ψ a i ψ(t) = H ψ(t) H = H - µ F(t) 0 ω F(t) = F sin(ωt) What happens if ω = E?

8 Resonances of the linear response function α(ω) = 1 [ 0 ˆµ n n ˆµ 0 ω n0 ω n>0 + 0 ˆµ n n ˆµ 0 ] ω n0 + ω

9 Absorption µ(t) fs 1.0 ρ B F(t) time ρ B (t) = 1 2 f ˆV ω 0 F ω 2 t 2

10 Rabi Oscillations 1 10 fs µ(t) ρb F(t) time

11 Relaxation S n ESA IC ps ISC S 1 ns-μs ESA IC ps T n OPA TPA Fluoresc. ns T 1 Phosphoresc. μs-ms S 0

12 Density matrix formalism The density operator is defined as ˆρ = s p(s) ψ s (t) ψ s (t), where the sum denoted a classical ensemble average. If all systems in the ensemble are identical then the summation contains a single term with unit probability. With wave function expansions in the form of projections onto the eigenstates ψ s (t) = cn(t) n, s n the matrix elements of the density operator becomes equal to ρ mn = s p(s) c s mc s n.

13 Coherence An ensemble of systems with equal state populations, but for which the phases of the wave function components varies in an incoherent manner, will have vanishing elements ρ mn (m n): ρ = 1 ( ) An ensemble that can be described by a single wave function, on the other hand, is fully coherent and is said to be in pure state: ψ = 1 ) (e iφa ψ a + e iφ b ψ b 2 with a density matrix: ρ = 1 ( 2 1 e i(φa φ ) b) e i(φ b φ a). 1

14 Time evolution and the Liouville equation Let us consider a pure state situation. Then t ˆρ = t ψ ψ + ψ t ψ = 1 [H, ˆρ], i which is known as the Liouville equation. For all intents and purposes, this equation provides an identical description of the QM system as if one chooses to work with the Schrödinger equation and a wave function approach. The difference lies in the possibility to modify the Liouville equation as to treat effects of spontaneous and collision-induced relaxation and heat bath interactions.

15 Liouville equation with relaxation t ρ mn = 1 i [Ĥ, ˆρ] mn γ mn (ρ mn ρ eq mn) The diagonal elements of the damping parameter matrix will govern the spontaneous population decays: τ n = 1/γ nn ; Γ n = γ nn, n 0; Γ 0 = 0 we can, for a pure state, draw a conclusion that the off-diagonal elements of the density matrix will depend on time according to ρ mn (t) = c m (t)c n(t) = c m (0)c n(0) e (Γm+Γn)t/2, and we must therefore have γ mn = (Γ m + Γ n )/2.

16 Time propagation of density matrix t ρ mn = 1 i [Ĥ, ˆρ] mn γ mn (ρ mn ρ eq mn) Initialization Time Propagation H=array([[Ea, 0], [0, Eb]]) mu=array([[0, muab], [muba, 0]]) F=Fw*sin(w*t)*erf rho_eq=array([[1,0], [0,0]]) rho[:,:,0]=[[1,0], [0,0]] for k in range(1,n): rho[:,:,k] = rho[:,:,k-1] - delta*1.0j*( dot(h - mu*f[k-1],rho[:,:,k-1]) - dot(rho[:,:,k-1],h - mu*f[k-1]) ) - delta*gamma*(rho[:,:,k-1]-rho_eq) P[k] = dot(rho[:,:,k],mu).trace() popa=rho[0,0,:] popb=rho[1,1,:]

17 1/Γ 10 fs 1.0 ρ b (t) 0.5 1/e ρ(0) = time ( ) 0 0 ; ρ eq = 0 1 ( ) 1 0 ; γ = Γ 0 0 ( 1 ) 1/2 1/2 1

18 Induced polarization in two-level atom (resonant field, relaxation) Γ = a.u. 10 fs µ(t) ρ b (t) F(t) time

19 µ(t) F(t) time α = i max[µ(t)] F ω 80i a.u.

20 Density matrix from perturbation theory with leads to From ρ (N) mn = e (iωmn+γmn)t t ρ mn (t) = ρ (0) mn + ρ (1) mn + ρ (2) mn +... ρ (0) mn = δ m0 δ n0 1 i [ ˆV, ˆρ (N 1) ] mn e (iωmn+γmn)t dt. ˆΩ (1) = Tr(ˆρ (1) ˆΩ) we identify the linear response function as [ ] ˆΩ; ˆV ω = 1 0 ˆΩ n n ˆV ω ˆV ω n n ˆΩ 0. ω n0 ω iγ n0 ω n0 + ω + iγ n0 n

21 Two-level atom: response function value Observable: ˆΩ = ˆµ Perturbation: ˆV ω = ˆµ E E = E b E a E b ψ b E a ψ a E = 0.5 a.u. µ ab = 1.0 a.u. ω = 0.5 a.u. γ = a.u. α(ω) = µ ab 2 E ω i γ + µ ab 2 E + ω + i γ µ ab 2 i γ = 80i

22 Complex Polarizability α( ω;ω) Real part Imag part FWHM = Γ ω

23 * * * * Pole structure Im(z) Re(z) * * * * Â; ˆV z = 1 n>0 [ ] 0  n n ˆV ˆV n n  0 ω n z iγ n ω n + z + iγ n Linear response function is convergent for all real frequencies (but will be complex). No poles in the upper half plane is a characteristic required for causal propagators.

24 Relaxation in wave function theory Apply the Ehrenfest theorem to ˆΩ = n m : t m ψ(t) ψ(t) n = 1 [ ] m Ĥ ψ(t) ψ(t) n m ψ(t) ψ(t) Ĥ n. i The above equation is thus a mere repetition of the Liouville equation for density matrix element ρ mn, and a suitable equation-of-motion with relaxation in wave function theory is: t ψ ˆΩ nm ψ = 1 [ ] i ψ [ˆΩ nm, Ĥ] ψ γ mn ψ ˆΩ nm ψ ψ eq ˆΩ nm ψ eq.

25 Linear response functions Molecular spectroscopies are interpreted as responses to electromagnetic fields or geometric perturbations. The linear responses of a molecular property to a perturbation are given by Â; ˆV ω = 1 n>0 [ 0  n n ˆV 0 + ω n ω iγ n0 ] 0 ˆV n n  0 ω n + ω + iγ n0 Spectroscopies and linear response functions Polarizability ˆµ; ˆµ ω Magnetizability ˆm; ˆm 0 Optical rotation ˆµ; ˆm ω Electronic circular dichroism ˆµ; ˆm ωf IR intensities ˆµ; Ĥ0/ R ω NMR parameters...

26 Nonlinear response functions The nonlinear responses of a molecular property to a perturbation are given by [ Â; ˆV, ˆV ω1,ω 2 = 1 0  n n ˆV ω 1 k k ω ˆV 2 0 P1,2 2 [ω k ω 2 iγ k0 ][ω n (ω 1 + ω n,k>0 2) iγ n0] 0 + ˆV ω 2 k k ˆV ω 1 n n  0 [ω k + ω 2 + iγ k0 ][ω n + (ω 1 + ω 2) + iγ + 0 ˆV ω1 n n  k k ˆV ] ω2 0 n0] [ω n + ω 1 + iγ n0][ω k ω 2 iγ k0 ] Spectroscopies and nonlinear response functions Second-harmonic generation ˆµ; ˆµ, ˆµ ω,ω Electro-optical Pockels effect ˆµ; ˆµ, ˆµ ω,0 Optical Rectification ˆµ; ˆµ, ˆµ ω, ω Faraday Rotation ˆµ; ˆµ, ˆm ω,0 Magnetic Circular Dichroism ˆµ; ˆµ, ˆm ωf,0 Raman intensities...

27 Work done by EM fields on particles A point charge in an EM field experiences the Lorentz force: F = qe + q (v B) The magnetic force is perpendicular to the magnetic field and the velocity (or current j = qv) of the particle. The latter implies that magnetic forces do not work. The work becomes W = r(tb ) r(t a) F dr = tb t a F vdt = tb t a qe vdt = tb t a E jdt Generalizing from a single moving charge to a current density, we get an instantaneous power : dw = (E j) dτ dt

28 Linear absorption cross section In the electric dipole approximations, the average change of absorbed energy per unit volume becomes d absorbed energy = E(t) j(t) dt volume T = N ω Im [α(ω)] I ε 0 c T where the angular brackets indicate the average over one period of oscillation T. We have used E(t) = E 0 (e iωt + e iωt ); P(t) = ε 0 E 0 (χe iωt + χ e iωt ); We identify the linear absorption cross section as σ(ω) = j(t) = P(t) ; I = 2ε 0 ce0 2 t ε 0 χ(ω) = N α(ω) ω Im [α(ω)] ε 0 c

29 Linear Optics The degree to which the external field F (t) manages to set the charges in motion is, to first order, expressed in terms of the linear electric polarizability α. In linear optics, the time-dependent polarization is described with an expression such as µ(t) = µ 0 + αf (t) where µ 0 is the permanent electric dipole moment of the molecule. The polarizability tensor is available from our theory in the form of a linear response function α αβ (ω) = ˆµ α ; ˆµ β ω

30 Nonlinear Optics In nonlinear optics, a generalization of the polarization is introduced in terms of a Taylor series in the electric field µ(t) = µ 0 + αf (t) βf 2 (t) γf 3 (t) + This equation introduces the first-order (nonlinear) hyperpolarizability β, the second-order hyperpolarizability γ, and so forth. Historical remarks: The name hyperpolarizability was given due to the presumed increase in the linear polarizability for systems in strong electric fields. (hyper: above) The name hypopolarizability was suggested for systems were there was a decrease in the linear polarizability, but it has not entered into common circulation. (hypo: under) C.A. Coulson, A. Macoll, L.E. Sutton, Trans. Faraday Soc. 48, 106 (1952).

31 Fourier decomposition of electric field F α (t) = ω F ω α e iωt This yields: µ α (t) = µ 0 α + ω α αβ ( ω; ω)f ω β e iωt + 1 β αβγ ( ω σ ; ω 1, ω 2 )F ω1 2 β F ω 2 γ e iωσt ω 1,ω γ αβγδ ( ω σ ; ω 1, ω 2, ω 3 )F ω1 6 β F ω 2 γ F ω 3 δ e iωσt + ω 1,ω 2,ω 3 where ω σ denotes the sum of optical frequencies: for terms involving β then ω σ = (ω 1 + ω 2 ) and for terms involving γ then ω σ = (ω 1 + ω 2 + ω 3 ). The Einstein summation convention for repeated subscripts is assumed here and elsewhere.

32 Tensor relations regarding frequency sign µ α (t) = µ 0 α + ω α αβ ( ω; ω)f ω β e iωt +... Since the molecular polarization µ(t) as well as the electric field F (t) are real, we have α(ω; ω) = [α( ω; ω)] β(ω σ ; ω 1, ω 2 ) = [β( ω σ ; ω 1, ω 2 )] γ(ω σ ; ω 1, ω 2, ω 3 ) = [γ( ω σ ; ω 1, ω 2, ω 3 )]

33 Tensor intrinsic symmetries ω 1,ω 2,ω 3 γ αβγδ ( ω σ ; ω 1, ω 2, ω 3 )F ω1 β F ω 2 γ F ω 3 δ e iωσt We note that any pairwise interchange of the indices and frequencies {β, ω 1 }, {γ, ω 2 }, and {δ, ω 3 } can be made without altering the physically observable polarization µ(t). It is therefore customary, but not necessary, to demand that the individual tensor elements are intrinsically symmetric β αβγ ( ω σ ; ω 1, ω 2 ) = β αγβ ( ω σ ; ω 2, ω 1 ) γ αβγδ ( ω σ ; ω 1, ω 2, ω 3 ) = γ αβδγ ( ω σ ; ω 1, ω 3, ω 2 ) = γ αγβδ ( ω σ ; ω 2, ω 1, ω 3 ) = γ αγδβ ( ω σ ; ω 2, ω 3, ω 1 ) = γ αδγβ ( ω σ ; ω 3, ω 2, ω 1 ) = γ αδβγ ( ω σ ; ω 3, ω 1, ω 2 )

34 Tensor overall symmetries In the nonresonant regions of the spectrum, the polarizabilities are real valued, and the tensors possess overall permutational symmetry: β αβγ ( ω σ ; ω 1, ω 2 ) = β βαγ (ω 1 ; ω σ, ω 2 ) γ αβγδ ( ω σ ; ω 1, ω 2, ω 3 ) = γ βαγδ (ω 1 ; ω σ, ω 2, ω 3 ) = γ γβαδ (ω 2 ; ω 1, ω σ, ω 3 ) = γ δβγα (ω 3 ; ω 1, ω 2, ω σ )

35 Example I: bichromatic laser Let two lasers A and B, which operate at frequencies ω A and ω B, respectively, interact. The external electric field experienced by the molecular system will in this case become F α (t) = F ω A α cos(ω A t) + F ω B α cos(ω B t) Including terms up to second-order in the field, the time-dependent polarization will be µ α (t) = µ 0 α + α αβ [F ω A β β αβγ[f ω A β cos(ω At) + F ω B β cos(ω At) + F ω B β cos(ω B t)] cos(ω B t)][f ω A γ cos(ω A t) + F ω B γ cos(ω B t)]

36 Induced polarization cos u cos v = 1 2 [cos(u + v) + cos(u v)] µ α (t) = µ 0 α + [ ααβ ( ω; ω)fβ ω cos(ωt) ω={ω A,ω B } β αβγ(0; ω, ω)fβ ω F γ ω + 1 ] 2 β αβγ( 2ω; ω, ω)fβ ω F γ ω cos(2ωt) + β αβγ ( (ω A + ω B ); ω A, ω B )F ω A β F ω B γ cos([ω A + ω B ]t) + β αβγ ( (ω A ω B ); ω A, ω B )F ω A β F ω B γ cos([ω A ω B ]t).

37 Nonlinear Optical Processes Second-harmonic generation: 2ω A and 2ω B Sum-Frequency Generation: (ω A + ω B ) Difference-Frequency Generation: (ω A ω B )

38 Wave particle duality Light can either be considered a classical electromagnetic wave or as a stream of photons and both pictures are useful to understand linear and nonlinear optical phenomena.

39 Difference-Frequency Generation E 1 ω B ω A (ω A ω B ) The light quanta of frequency ω A is annihilated and those of frequencies ω B and (ω A ω B ) are both created (two-photon emission). The creation of the photon with frequency ω B is a result of stimulated emission induced by laser B. It is thus in principle possible to amplify a weak light signal B with a pump laser A. 0

40 Example II: monochromatic laser and static field F α (t) = F 0 α + F ω α cos(ωt) µ α (t) = µ 0 α + α αβ [F 0 β + F ω β cos(ωt)] β αβγ[f 0 β + F ω β cos(ωt)][f 0 γ + F ω γ cos(ωt)] γ αβγδ[f 0 β + F ω β cos(ωt)][f 0 γ + F ω γ cos(ωt)][f 0 δ + F ω δ cos(ωt)]. Re-write on form: µ α (t) = µ 0 α + µ ω α cos(ωt) + µ 2ω α We then get... cos(2ωt) + µ 3ω α cos(3ωt),

41 µ 0 α = µ 0 α + α αβ (0; 0)F 0 β β αβγ(0; 0, 0)F 0 βf 0 γ γ αβγδ(0; 0, 0, 0)F 0 βf 0 γf 0 δ β αβγ(0; ω, ω)f ω β F ω γ γ αβγδ(0; ω, ω, 0)F 0 βf ω γ F ω δ, µ ω α = α αβ ( ω; ω)f ω β + β αβγ ( ω; ω, 0)F ω β F 0 γ γ αβγδ( ω; ω, 0, 0)F ω β F 0 γf 0 δ γ αβγδ( ω; ω, ω, ω, )F ω β F ω γ F ω δ, µ 2ω α = 1 4 β αβγ( 2ω; ω, ω)f ω β F ω γ γ αβγδ( 2ω; ω, ω, 0)F ω β F ω γ F 0 δ, µ 3ω α = 1 24 γ αβγδ( 3ω; ω, ω, ω)f ω β F ω γ F ω δ.

42 We prefer Taylor expansions of the polarization µ 2ω α = 1 4 β αβγ( 2ω; ω, ω)f ω β F ω γ γ αβγδ( 2ω; ω, ω, 0)F ω β F ω γ F 0 δ An alternative summary of this result is sometimes expressed as an expansion of the polarization amplitudes in terms of the electric field amplitudes µ ωσ α = α( ω σ ; ω 1 )F ω K (2) β( ω σ ; ω 1, ω 2 )F ω 1 F ω K (3) γ( ω σ ; ω 1, ω 2, ω 3 )F ω 1 F ω 2 F ω 3 +, where the factors K (n) are required for the polarization related to the molecular response of order n to have the same static limit.

43 Conversion factors Process Frequencies Factor Second-order processes K (2) Static 0; 0, 0 1 EOPE a ω; ω, 0 2 SHG b 2ω; ω, ω 1/2 Third-order processes K (3) Static 0; 0, 0, 0 1 EOKE c ω; ω, 0, 0 3 IDRI d ω; ω, ω, ω 3/4 ESHG e 2ω; ω, ω, 0 3/2 THG f 3ω; ω, ω, ω 1/4 a Electro-optical Pockels effect. b Second-harmonic generation. c Electro-optical Kerr effect. d Intensity-dependent refractive index. e Electric field-induced second harmonic generation. f Third-harmonic generation.

44 Dispersion of polarizability α αβ ( ω; ω) = 1 [ 0 ˆµ α n n ˆµ β 0 ω n0 ω n + 0 ˆµ ] β n n ˆµ α 0 ω n0 + ω In the nonrelativistic limit, the unperturbed eigenfunctions of the molecular Hamiltonian can be chosen as real and the numerators of the two terms are then equal and can be separated out: α αβ ( ω; ω) = 1 With use of the Taylor expansion n 2 0 ˆµ α n n ˆµ β 0 ω n0 1 (ω/ω n0 ) 2 we get x = x k for x < 1, k=0

45 Cauchy moment expansion α αβ ( ω; ω) = 1 ω 2k S( 2k 2) for ω/ω on < 1, k=0 where the Cauchy moments S( k) have been introduced as S( k) = n f αβ n0 ωn0 k ; f αβ n0 = 2ω n0 0 µ α n n µ β 0. The Cauchy moments are independent of the optical frequency. The expansion of the polarizability in the optical frequency contains only terms of even power. The Cauchy moments with α = β are positve, the dispersions of the diagonal components of the polarizability are bound to be positive.

46 Dispersion of hyperpolarizability The dispersions of hyperpolarizabilities are considerably more complicated, e.g., β αβγ ( ω σ ; ω 1, ω 2 ) = 1 0 ˆµ α n n ˆµ β p p ˆµ γ 0 P σ,1,2 2 (ω n0 ω σ )(ω p0 ω 2 ) where ˆµ is the fluctuation dipole moment operator ˆµ 0 ˆµ 0. If we consider a diagonal component of the β-tensor, it is clear that the frequency dependence rests in [ ] 1 P σ,1,2 (ω n0 ω σ )(ω p0 ω 2 ) 1 ( ) k ( ) l = P σ,1,2 ωσ ω2 ω n0 ω p0 n,p k,l=0 ω n0 ω p0

47 Auxiliary permutation formulas 1 P σ,1,2 ω n0ω p0 k,l=0 ( ωσ ) k ( ) l ω2 ω n0 ω p0 We introduce the notaion of ω σ = ω σ as well as P σ,1,2,...,n = P, and get: 1 = (n + 1)!, P ω 1 = n! P ω1 2 = n! P n ω k = 0, k= σ n ωk 2 = n! ωl, 2 k= σ ω 1ω 2 = (n 1)! P n ω k ω l = (n 1)! ωl, 2 k= σ l k where ω 2 L is defined by the above equation.

48 General dispersion formula for first hyperpolarizability β zzz ( ω σ ; ω 1, ω 2 ) = β zzz (0; 0, 0) + A ω 2 L +... where the coefficient A has been introduced as A = n,p T zzz ( np ω n0 ω p0 2 ω 2 n0 + 2 ωp ω n0 ω p0 ), with T αβγ np = 0 ˆµ α n n ˆµ β p p ˆµ γ 0.

49 First hyperpolarizability of hydrogen fluoride β zzz ( 2ω;ω,ω) β zzz ( ω;ω,0) β zzz ω 2 L -8.5 β zzz ω Hartree Fock/t-aug-cc-pVTZ results for SHG and EOPE optical processes. The dipole moment is directed along the positive z-axis and the experimental bond length of a.u. is employed. All quantities are given in a.u.

50 General dispersion formula for second hyperpolarizability γ αβγδ ( ω σ; ω 1, ω 2, ω 3 ) = 1 P σ,1,2,3 3 0 ˆµ α n n ˆµ β m m ˆµ γ p p ˆµ δ 0 nmp (ω n0 ω σ)(ω m0 ω 2 ω 3 )(ω p0 ω 3 ) 0 ˆµ α n n ˆµ β 0 0 ˆµ γ m m ˆµ δ 0 nm (ω n0 ω σ)(ω m0 ω 3 )(ω m0 + ω 2 ) γ zzzz( ω σ; ω 1, ω 2, ω 3) = γ zzzz(0; 0, 0, 0) + A ω 2 L where the frequency-independent coefficient A is equal to A = n,m,p n,m T zzzz nmp ω n0 ω m0 ω p0 T zzzz nm ω n0 ω 2 m0 ( 6 ωn ωm0 2 ), ( 6 ω 2 n ω 2 m0 + 6 ωp ω n0 ω m0 + 2 ω n0 ω p0 + 4 ω m0 ω p0 ) with T αβγδ nmp = 0 ˆµ α n n ˆµ β m m ˆµ γ p p ˆµ δ 0, T αβγδ nm = 0 ˆµ α n n ˆµ β 0 0 ˆµ γ m m ˆµ δ 0.

51 Second hyperpolarizability of hydrogen fluoride γ zzzz γ zzzz γ zzzz ( 3ω;ω,ω,ω) γ zzzz ( 2ω;ω,ω,0) γ zzzz ( ω;ω, ω,ω) γ zzzz ( ω;ω,0,0) ω 2 L ω Hartree Fock/t-aug-cc-pVTZ results for THG, ESHG, IDRI, and EOKE optical processes. The dipole moment is directed along the positive z-axis and the expt bond length of a.u. is employed. All quantities are given in a.u.

52 Summary In nonresonant spectral regions, polarizabilities are real and determine scattering of radiation In near-resonant and resonant spectral regions, polarizabilities are complex with imaginary parts that determine absorption of radiation An overall sign change of optical frequencies corresponds to a complex conjugation of the polarizability tensor In nonresonant regions, tensors are overall symmetric. General dispersion formulas can be derived showing that diagonal tensor elements and isotropic averages have positive dispersion