Generalized electronic diabatic approach

O. Tapia, Quantum linear superposition theory for chemical processes A generalized electronic diabatic approach, Adv. Quantum Chem. 56 (2009) 31-93. 

A manner to do away with the problem is to introduce appropriate algorithms in the sense that mappings from real space to Hilbert space can be defined. The generalized electronic diabatic, GED approach fulfils this constraint while the BO scheme as given by Meyer [2] does not due to an early introduction of center-of-mass coordinates and rotating frame. The standard BO takes a typical molecule as an object description. Similarly, the wave function is taken to describe the electrons and nuclei. Thus, the adiabatic picture follows. The electrons instantaneously follow the position of the nuclei. This picture requires the system to be always in the ground state. 

We examine a post-Bom-Oppenheimer approach based on a generalized electronic diabatic (GED) ansatz for electronuclear dynamics in external electromagnetic belds. The model is a quantum -electron system interacting with a set of m classical positive charges located at = ( 1) m) itt laboratory space. The system s Coulomb Hamiltonian is Wc =... 

The GED approach is a general procedure based on the exact solutions to the n-electron system. Only one Hamiltonian is required at variance with the infinite Hamiltonian approach (defined on the parametric -space) characteristic of the BO scheme. All the base functions are expanded from a unique origin of the I-frame. The characteristics of the n-electrons diabatic base functions are independent from the positions taken by the sources of the external potential. 

The correlation function can be written in a basis set Ra) = R) a) chosen as the tensor product of the coordinate representation for the nuclear degrees of freedom, and a nuclear coordinate independent electronic basis. In this chapter we shall refer to this electronic representation as the diabatic basis. (We refer the interested reader to reference [40] for the development of this approach with a more general electronic representation). By inserting resolutions of the identity in this basis, one obtains... 

Non-adiabatic case. The diabatic states or the adiabatic states may be used to construct the basis set for the motion of the electrons and nuclei in the non-adiabatic approach. Such a basis function is taken as a product of the electronic (diabatic or adiabatic) wave function and of a rovibrational wave function that depends on R. In a non-adiabatic approach the total wave function is a superposition of these product-like contributions [a generalization of eq. (6.7)] ... 

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

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