Dancoff

In order to take into account the increased complexity of molecules with heteroatoms (i.e., graphs with coloured vertices), Bertz combined C(n) with the information for chemical composition, Icc, introduced by Branson 84a) and by Dancoff and Quastler 84b), resulting in ... 

Univ. Illinois Press, Urbana 1953 b) Dancoff, A., Quastler, H. ibid. 

A different analysis applies to the LR approach (in either Tamm-Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. 

R. Cammi, B. Mennucci, J. Tomasi, Fast evaluation of geometries and properties of excited molecules in solution A Tamm-Dancoff Model with application to 4-dimethylaminobenzonitrile. J. Phys. Chem. A 104, 5631 (2000)... 

Tamm-Dancoff Green s function results relative intensities given in parentheses. c Adiabatic ionization energies. 

The matrix Bis numerically much less important than A and accordingly y << x. This suggests a simplification of the foregoing equations by setting B = 0. This Tamm-Dancoff approximation [11] leads to a symmetric matrix eigenvalue equation... 

Since DFT has essentially the same mean-field formalism as the HF theory and share much the same computational algorithm, it is not surprising that it has the excited-state counterparts corresponding to TDHF and CIS. They—TDDFT [83-88] and Tamm-Dancoff TDDFT [89], respectively - can be derived analogously to Section 2.2.1 with the only differences being in the definitions of the operator (now called the Kohn-Sham or KS Hamiltonian) and its derivative with respect to the density matrix [see Eq. (2-7)]. The latter is... 

It may not be obvious that the self interaction problem is present in TDDFT, but it can be made evident by comparing CIS and Tamm-Dancoff TDDFT for an excitation describable by just a pair of orbitals (xai = 1) ... 

The extension of TDDFT and Tamm-Dancoff TDDFT to crystalline polymers is straightforward within the formalisms of Section 2.2.2. Figure 2-7 summarizes the results of TDDFT calculations of the photoconduction, photoemission, and optical absorption thresholds (energy gap, ionization energy, and excitation energy) of polyethylene as a function of basis set [50], The Slater-Vosko-Wilk-Nusair functional [116, 117] is used, but the following conclusion is unaltered... 

The corrected Linear Response approach (cLR) consists in the use the TDDFT relaxed density and the corresponding apparent charges (7-38) into Eqs. (7-36) and (7-37) to obtain the first-order approximation to the state specific free energy of the excited state. The details of the implementation are described in Ref. [17], This corrected Linear Response computational scheme can be applied to the analogous of the Time Dependent Hartree-Fock approach either in the complete (Random Phase Approximation) or approximated (Tamm-Dancoff approximation or Cl singles, CIS) version. 

As it is not possible to obtain TDDFT-SS results, the results refer to CIS method. In fact, this method can be obtained from two points of view one is to consider the method as a standard Cl, in which the wave function of the excited state is constructed by single excitations from the HF determinant and thus a SS solvent response can be obtained the other is to consider CIS as the result of the Tamm-Dancoff approximation applied to the linear response equation based on the HF wave function. The two ways of looking at the CIS method give the same equations in vacuo, but, as discussed above, they differ for molecules in solution due to the nature of the effective Hamiltonian. 

It is important to note, that for the accurate estimation of spectral intensities of the theoretical spectra it is not sufficient to consider only the lowest order processes. A sufficiently accurate yet relative simple computational scheme is the two-particle-hole Tamm-Dancoff approximation [36]. E>ue to the difficulties and complications involved in the computations there are only few such studies [32-35]. The majority of the authors use a simplified model for the estimation of the ionization energies, based on one of the following approximations ... 

Q [Shannon, 1948] [Dancoff and Quastler, 1953] [Rashevsky, 1960] [Bonchev and Kamenska, 1978] [Kier, 1980c] [Rouvray, 1997] [Agrafiotis, 1997]... 

Dancoff, S.M. and Quastler, H. (1953). Essays on the Use of Information Theory in Biology. University of Illinois, Urbana (ILL). 

J. Hutter (2003) Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework. J. Chem,. Phys. 118, p. 3928... 

FOCI - first-order configuration interaction. Outer valence Green s function. Extended two-particle-hole Tamm-Dancoff approximation. 

Obviously, those are the same considerations as we went through in order to obtain Eqs (90) and (91) and the electron propagator method and the ADC are thus equivalent methods. Using n = 2 in Eq. (93) we determine and n — 3 gives The U matrix in Eq. (93) corresponds to the transition matrix (cf. Eq. (75)). Both and only contain C and D terms (see Eqs (87), (88), (90) and (91), i.e. hj = hj alone. From Eq. (63) we see that we may classify the operators in hj as 2p-lh (two-particle, one-hole) and 2h-lp operators, and the n = 3 ADC approach, corresponding to the third-order electron propagator method, is therefore referred to as the extended 2p-lh Tamm-Dancoff approximation (TDA) (Walter and Schirmer, 1981). A fourth-order approximation to the ADC equations has also been described (Schirmer et ai, 1983) but not yet tested in actual applications. 

AGPTDA antisyrametrized geminal power Tamm-Dancoff approximation (Section VII. C),... 

Liu J, Liang W (2011) Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation. J Chem Phys 135 014113... 

---