Exact linear response theory

6 Time-dependent theory and linear response 6.4.2 Exact linear response theory 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4 i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA / i(r) i / (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih pi(rt) = G(rt)4 i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

For t 0, a weak perturbing potential Av(rt) induces a screening potential such that AQ = Av + AQS. The first-order perturbation equations are 

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This result cannot be reconciled with that of Dirac [13] and with the structure of the second-quantized Hamiltonian [12,14]. The implication is that the locality hypothesis fails for Ex and probably for Ec, which can be incorporated in a formally exact extension of Dirac s derivation. [17] Thus restriction to local exchange and correlation potentials is inconsistent with exact linear-response theory. 

Exact linear-response theory based on TDOFT is a potentially powerful methodology for treating excitations and polarization response. 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

Nesbet, R.K. (1999). Exact exchange in linear-response theory, Phys. Rev. A 60, R3343-R3346. 

The first order effect of L (t) is obtained exactly as in linear response theory and generates the counterpart of terms in (Aa) in eqs 11 and 12. The new problem comes in obtaining the second order effect of the dipole torque operator L (t) for terms in E, as one must then evaluate the consequences of operations of the form... 

CCSD(T) method. The question then naturally arises as to how these methods can be extended to excited states. For the iterative methods, the extension is straightforward by analyzing the correspondence between terms in the CC equations and in H, one can define an H matrix for these methods, even though it is not exactly of the form of a similarity-transformed Hamiltonian. If one follows the linear-response approach, one arrives at the same matrix in the linear response theory, one starts from the CC equations, rather than the CC wave function, and no CC wave function is assumed. This matrix also arises in the equations for derivatives of CC amplitudes. In linear response theory, this matrix is sometimes called the Jacobian [19]. The upshot is that excited states for methods such as CCSDT-1, CCSDT-2, CCSDT-3, and CC3 can be obtained by solving eigenvalue equations in a manner similar to those for methods such as CCSD and CCSDT. 

In this section we develop a general linear response theory for which the coupling of the system to the field in Eqs. [91] is small [i.e., fjt) 1]. Linear response theory should, in general, take into account the possibility of phase space compression coming from either the coupling to the external field or from the extended phase space variables. In traditional formulations of linear response theory, phase space compression has not been carefully considered. The present formulation treats the first level of compressibility exactly, for example, the compressibility due to the presence of an extended system (such as ther-... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The familiar shear modulus of linear response theory describes thermodynamic stress fluctuations in equilibrium, and is obtained from (5b, lid) by setting y = 0 [1, 3, 57], While (5b) then gives the exact Green-Kubo relation, the approximation (lid) turns into the well-studied MCT formula (see (17)). For finite shear rates, (lid) describes how afflne particle motion causes stress fluctuations to explore shorter and shorter length scales. There the effective forces, as measured by the gradient of the direct correlation function, = nc = ndck/dk, become smaller, and vanish asympotically, 0 the direct correlation function is connected... 

The failure of the DFT linear-response theory to reduce to the exact formalism of Dirac [13] in the exchange-only limit [12] is symptomatic of the inadequacy of the locality hypothesis. It will be shown here that on dropping this hypothesis a linear-response theory can be derived that is formally exact for both exchange and correlation. As will be discussed in more detail in the following Section, an exact but implicit orbital functional exists for the correlation energy Ec [31]. This produces a formally exact correlation term in the OEL equations, defined by the orbital functional derivative... 

By using linear-response theories, dispersion interactions can be calculated directly in the framework of the Kohn-Sham method. The adiabatic connection/fluctuation-dissipation theorem (AC/FDT) method is a linear-response theory for exactly calculating dispersion interactions within the framework of the Kohn-Sham method (Langreth and Perdew 1975). In this AC/FDT method, electron correlation is calculated as the energy response quantity for the spontaneous fluctuations of electronic motions coming from the perturbation of the interelectronic interactions, as follows ... 

In general theory, the first-order time-dependent density can be calculated from the exact linear response function x acting on an external perturbing time-dependent potential Vi(r, f) ... 

Both transition energies and oscillator strengths are needed for determination of optically allowed absorption spectra. In the multi-configuration version of the linear response theory (MCLR) one constructs an approximation to the exact linear response function by exposing the optimized (MC) SCF wavefunction 0> to a time-dependent perturbation. In this case the time-dependent wave function assumes the form... 

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