Density matrix-based energy functional

As discussed, for a formulation of SCF theories suitable for large molecules, it is necessary to avoid the nonlocal MO coefficient matrix, which is conventionally obtained by diagonalizing the Fock matrix. Instead we employ the one-particle density matrix throughout. For achieving such a reformulation of SCF theory in a density matrix-based way, we can start by looking at SCF theory from a slightly different viewpoint. To solve the SCF problem, we need to minimize the energy functional of... 

In the derivation of density matrix-based SCF theory below, we do not employ the chemical potential introduced by LNV, but instead we follow the derivation of Ochsenfeld and Head-Gordon, because McWeeny s purification automatically preserves the electron number.Therefore, to avoid the diagonalization within the SCF procedure, we minimize the energy functional... 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

Compared to computational approaches based on the g-electron density, approaches based on the g-electron reduced density matrix have the advantage that the kinetic energy functional can be written in an explicit form ... 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

In this section we are going to develop a different approach to the calculation of excitation energies which is based on TDDFT [69, 84, 152]. Similar ideas were recently proposed by Casida [223] on the basis of the one-particle density matrix. To extract excitation energies from TDDFT we exploit the fact that the frequency-dependent linear density response of a finite system has discrete poles at the excitation energies of the unperturbed system. The idea is to use the formally exact representation (156) of the linear density response n j (r, cu), to calculate the shift of the Kohn-Sham orbital energy differences coj (which are the poles of the Kohn-Sham response function) towards the true excitation energies Sl in a systematic fashion. 

Most importantly, these systems are amenable to the Electron Localization Function (ELF) method [21]. This is a local measure based on the reduced second-order density matrix, which as pioneered by Lennard-Jones [22] should retain the chemical significance and at the same time reduce the complexity of the information contained in the square of the wave function ELF is defined in terms of the excess of local kinetic energy density due to the Pauli exclusion principle, T r), and the Thomas-Fermi kinetic energy density, Th(r) ... 

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