Response function symmetry

In the earlier sections of this chapter we reviewed the many-electron formulation of the symmetry-adapted perturbation theory of two-body interactions. As we saw, all physically important contributions to the potential could be identified and computed separately. We follow the same program for the three-body forces and discuss a triple perturbation theory for interactions in trimers. We show how the pure three-body effects can be separated out and give working equations for the components in terms of molecular integrals and linear and quadratic response functions. These formulas have a clear, partly classical, partly quantum mechanical interpretation. The exchange terms are also classified for the explicit orbital formulas we refer to Ref. (302). 

It is important to note that the two electric fields that lead to a Raman transition can have different polarizations. Information about how the transition probability is affected by these polarizations is contained within the elements of the many-body polarizability tensor. Since all of the Raman spectroscopies considered here involve two Raman transitions, we must consider the effects of four polarizations overall. In time-domain experiments we are thus interested in the symmetry properties of the third-order response function, R (or equivalently in frequency-domain experiments... 

As is well-known, the critical point can be found from locating the divergence of the response function ( susceptibility ) S = dm/dH, which signals the onset of symmetry breaking and spontaneous order. The simplest case occurs for Hj=0, and then... 

Besides that, the response functions Xs and x satisfy the symmetry relations [106]... 

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. 

Even though Eqs. (135) and (137) have a different structure they give numerically identical results. This follows from the permutation symmetry of the response function. In this case there is no computational advantage to use one expression before the other, we have to solve the same set of response equations in both cases. 

It is less obvious to what extent the measured response Hx can be permuted. This, however, is apparent from the left hand side of (24) where one has complete permutation symmetry in the denominator. A more complete discussion of symmetry of response functions is found in Christiansen et al. [13]. 

If the perturbation operators Ha and Hb have the same symmetry with respect to time reversal, the corresponding linear response function is real... 

S ii(q = 0) describes the critical scattering along the coexistence curve, while Scoii(q = 0) describes the critical scattering for T > Tc and A i = 0 (i.e., at critical composition < > = crit). This fact that for the description of the response function, different expressions need to be used above and below the phase transition, in order to take into account the spontaneous symmetry breaking that appears in the thermodynamic limit between the coexisting phases, is well known [105] but has not been considered in some of the work on polymer mixtures [101-103, 107]. 

Since S(q ) has such a smooth temperature variation in the simulation, Fig. 47a, Fried and Binder [325] rely on dynamic criteria for locating the MST, motivated by the experimental finding that the MST shows up most clearly in the dynamic response of the blockcopolymer melt (e.g. by an abrupt change in the frequency dependence of viscoelastic response functions at Tmst [317-323]. The idea is that in the lamellar phase there will be a spontaneous symmetry breaking, S(q) depends on the direction of q for T < Tmst, since the orientation perpendicular to the lamellar is singled out, while for T > Tmst the symmetry of the disordered phase requires that S(q) is spherically symmetric. However, this... 

The response function of Eq. (96) has poles at the orbital energy differences j — i) (where one of the orbitals xpj and xpt is occupied and the other empty). Therefore, these poles represent first approximations to the excitation energies. Better estimates can be obtained by identifying the poles of the true response function. However, the resulting equations are very involved and have been solved for only few systems like, e.g., closed-shell atoms where the spherical symmetry and closed shells lead to important simplifications. In order to demonstrate the... 

Over recent years there has been a steady growth of interest in vibrational effects in the context of ab initio calculations of linear and non-linear molecular response functions. It has been realized that in some cases vibrational may rival electronic contributions to the parameters controlling non-linear optical responses. This is particularly likely where the molecule is of higher symmetry (quadrupolar or octupolar rather than dipolar), and for lower frequency effects where there is little pre-resonant enhancement of the electronic contribution. The main features of the theoretical methodology for the calculation of vibrational response functions were established several years ago and the fundamental papers were reviewed in the previous volume. Recent developments have been the introduction of field induced co-ordinates, improved integration techniques and the first relativistic studies. ... 

The asymptotic low-fiequency behavior of the electronic Raman scattering response function, Im f.,y((y 0), for a d 2 2 superconductor in different symmetries ... 

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

The assumption, known as Kleinman symmetry, that the quadratic (and higher-order) response function is symmetric imder interchange of any pair of operators is only true at zero frequency, as is easily verified from O Eq. 5.31. Note that the quadratic response function is symmetric when permuting the operators B and C at co = co, which is the case for, for example. Second Harmonic Generation j jjk -2(o-,co,(o) = Pikj -2(o-,(o,co). Kleinman symmetry is often assumed in calculations of the electric dipole hyperpolarizabUity at low frequencies where it is approximately vahd. This reduces the number of independent tensor elements and thus the computational effort... 

Olsen and Jorgensen (1985, 1995) have derived and discussed response functions for exact, HF, and MCSCF wave functions in great detail, while Koch and Jorgensen (1990) presented a derivation for CC wave functions. The latter was modified by Pedersen and Koch (1997) to ensure proper symmetry of the response functions. Christiansen et al. (1998) have presented a derivation of dynamic response functions for variational as well as non-variational wave functions that resembles the way in which static response functions are deduced from energy derivatives. Linear and higher order response functions based on DFT have been presented by Salek et al. (2002). Damped response theory has been discussed by Norman et al. (2001) in the context of HF and MCSCF response theory. Nonpertur-bative calculations of static magnetic properties at the HF level have been presented by Tellgren et al. (2008, 2009). 

For time-dependent oscillatory perturbations, the response functions can be related to the derivatives of the time-averaged quasienergy (Christiansen et al. 1998 Saue 2002). The derivation, based on a time-dependent variational principle and the time-dependent HeUman-Feynman theorem, shows explicitly the symmetry of the response functions with respect to the permutation of aU the operators. 

The general definition of the quadratic response function, O Eq. 11.68, indicates its symmetry with respect to permutation of operators. Thus, for all the dipole hyperpolarizabilities we have... 

For atomic systems, theory can determine rather accurately the absolute shielding constants. First, fairly large nonelectronic contributions such as the zero-point vibrational corrections are absent. Secondly, because of symmetry only the diamagnetic term - an expectation value, which is much easier to compute than an accurate value of the response function - contributes. For instance, for the hydrogen atom the nonrelativistic shielding constant can be calculated as ... 

The third-order response can be divided in an instantaneous response due to the electronic hyperpolarizability plus a second noninstantaneous response defined by the nuclear dynamics and hence by the molecular dynamics. This second part includes all the relevant dynamic information in a complex liquid. The proper definition of the nuclear response function will be introduced later, here we recall only the symmetry rules that must be verified by it. In an isotropic medium, as a liquid or a glass, is subject to several symmetry rules [41]. According to... 

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