Excitations from Linear-Response Theory

The first self-consistent solution of the linear response (see 4.69) was performed by Zangwill and Soven in 1980 using the LDA/ALDA [42]. Their results for the photo-absorption spectrum of xenon for energies just above the ionization threshold are shown in Fig. 4.3. Once more the theoretical curve compares very well to experiments. 

Unfortunately, a full solution of (4.69) is still quite difficult numerically. Besides the large effort required to solve the integral equation, we need the non-interacting response function as an input. To obtain this quantity it is usually necessary to perform a summation over all states, both occupied and unoccupied [cf. (4.63)]. Such summations are sometimes slowly convergent and require the inclusion of many unoccupied states. There are however approximate frameworks that circumvent the solution of (4.69). The one we will present in the following was proposed by Petersilka et al. [22]. 

The density response function can be written in the Lehmann representation 

As the external potential does not have any special pole structure as a function of Lo, (4.61) implies that also n r,u)) has poles at the excitation energies, 17. On the other hand, xks has poles at the excitation energies of the non-interacting system, i.e. at the Kohn-Sham orbital energy differences e,-e,[cf.(4.63)].  

By rearranging the terms in (4.68) we obtain the fairly suggestive equation 

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We observe at this point that Eq. (S.S9) supplemented by Eq. (S.S8) expresses the most recent analytical result obtained to account for the effects of nonlinear excitation. Note, however, that the perturbation approach behind this equation means that it is unable to account for large deviations from linear response theory. In other words, both the intrinsically nonlinear statistics of the system under study and the intensity of the external excitation have to be assumed to be quite small. The rotational coimterpart of Eq. (S.S8) (< is replaced by the angular velocity comparison with the results obtained by applying the continued fraction procedure (CPF) (see Chapters III and IV). It has been shown that the deviation of the linear response theory from the CFP is intermediate between that predicted by Eq. (5.59) and that based on Suzuki s mean held approximation (Chapter V). (In agreement with the CFP, however, both predict that the decay of becomes slower with increases in the excitation parameter r = ( )exc/ " )eq -1.)... 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

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. 

To date, most applications of TDDFT fall in the regime of linear response. The linear response limit of time-dependent density functional theory will be discussed in Sect. 5.1. After that, in Sect. 5.2, we shall describe the density-functional calculation of higher orders of the density response. For practical applications, approximations of the time-dependent xc potential are needed. In Sect. 6 we shall describe in detail the construction of such approximate functionals. Some exact constraints, which serve as guidelines in the construction, will also be derived in this section. Finally, in Sects. 7 and 8, we will discuss applications of TDDFT within and beyond the perturbative regime. Apart from linear response calculations of the photoabsorbtion spectrum (Sect. 7.1) which, by now, is a mature and widely applied subject, we also describe some very recent developments such as the density functional calculation of excitation energies (Sect. 7.2), van der Waals forces (Sect. 7.3) and atoms in superintense laser pulses (Sect. 8). 

Equation (11) allows to interpret the electronic excitation in the Auger process as the medium response to a dynamic fluctuation of charge between the states i(r) and

linear response theory to an external pertirrbation which is the electrostatic potential v i(r) created by such charge fluctuation. In general terms, the response function r, m) completely determines the behavior of the system in response to an external perturbation, provided that the latter is sufficiently small and thus linear theory is applicable. The imaginary part of contains... 

New calculations of the various contributions to the second-order stopping power of a uniform FEG coming from the excitation of e-h pairs and plasmons have been reported. We have found that the equipartition rule, vahd within first-order perturbation (linear-response) theory, cannot be extended to higher orders in the external perturbation. We have also found that contributions from collective excitations to the Zj term are small. 

Another class of method tries to generate MR states of interest via the action of an excitation operator on a simple base function, usually of the ground state. The linear response-based theories based on CC reference functions have been proposed quite some time ago to achieve this goal, starting from the HF or ROHF ground reference state [22-25]. An interesting variant to handle non-trivial open-shell states via a spin-flip operator [26] has revived the interest in the generalization of the CC-based linear response theory to open-shell states. 

Figure 4 shows the results of exact DFT calculations for the He atom. On the left side of the diagram, we consider just transitions from the exact ground-state KS occupied orbital (Is) to unoccupied orbitals. These are not the true excitations of the system, nor are they supposed to be. However, applying TDDFT linear response theory, using the exact kernel with the exact orbitals, yields the exact excitation frequencies of the He atom. Spin decomposing produces both singlet and triplet excitations. 

The genesis of the failure of TDDFT can be traced to the fact that TDDFT is a linear response theory. When an excitation moves charge from one area in a molecule to another, both ends of that molecule will geometrically relax. While charge transfer between molecules can be well approximated by ground-state density functional calculations of the total energies of the species involved, TDDFT must deduce the correct transitions by... 

An alternative approach is based on the time-dependent density functional theory [40]. From the linear response theory, it can be shown that proper treatment of the excited states can be obtained from the solutions of a non-Hermitian eigenvalue problem [41],... 

To obtain excitation energies and properties within the time-dependent Kohn-Sham framework, it is possible to propagate in time the time-dependent electron density, through the solution of Eq. (4.60), and then extract energies and oscillator strengths from a Fourier analysis of the results [98-102]. Alternatively, the excited-state properties can be determined through the linear response theory. This is an efficient approach which avoids the direct solution of the time-dependent Kohn-Sham equations and is often used in practical applications. 

It is evident that if one drops the excitation operator, YjP from the working equations of MR-CEPALRT they naturally reduce to the corresponding ground state CEPA theories, SS-MRCEPA. This implies that SS-MRCEPA are the completely unperturbed versions of MR-CEPALRT, which is why we coin the term, MR-CEPALRT, for the linear response theory based on the SS-MRCEPA method. We not only get the excited state energies but also the ground state energy by solving the MR-CEPALRT equations. 

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

Within the QM continuum solvation framework, as in the case of isolated molecules, it is practice to compute the excitation energies with two different approaches the state-specific (SS) method and the linear-response (LR) method. The former has a long tradition [10-24], starting from the pioneering paper by Yomosa in 1974 [10], and it is related to the classical theory of solvatochromic effects the latter has been introduced few years ago in connection with the development of the LR theory for continuum solvation models [25-31],... 

Using the formalism of response theory [24,31], the scalar rotatory strength for a transition from the ground state 0) to an excited state n) can be evaluated as the residue of the linear response function. In the velocity gauge formulation, nR is given by the equation... 

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