Diabatic mixing

There are three ways of implementing the GP boundary condition. These are (1) to expand the wave function in terms of basis functions that themselves satisfy the GP boundary condition [16] (2) to use the vector-potential approach of Mead and Truhlar [6,64] and (3) to convert to an approximately diabatic representation [3, 52, 65, 66], where the effect of the GP is included exactly through the adiabatic-diabatic mixing angle. Of these, (1) is probably the most... 

The other techniques can be divided into a few categories (see also Ref. [14c]). One of them goes back to the idea first coined by Mulliken [41], Hush [42] and Lichten [8] of using molecular properties to determine the diabatic basis and is actually based on the Werner-Meyer-Macfas-Riera formula [32,33] for the adiabatic-to-diabatic mixing angle in terms of electronic dipole moments. This... 

If all four of the new basis states are included, the adiabatic mixed electronic/proton vibrational states are exactly the same as those obtained with the original four VB states. Moreover, the diabatic mixed electronic/proton vibrational states for the new basis states are exactly the same as the adiabatic states obtained with the settings Ho)ia,2a = ( o)ia,26 = ( o)i6,2a = ( )i6,26 = (as described at the end of the previous section). 

The adiabatic-diabatic mixing angle a is a complicated function of the Jacobi coordinates and an evaluation of A in these coordinates is cumbersome. This can be better accomplished in the E X e)-JT coordinates pjT and y. The latter are identified as the radius of the JT displacement and the pseudo-rotation angle (introduced above), respectively. These coordinates can be expressed in terms of the dimensionless normal coordinates (cartesian) of the e-type vibration Qx and Qy) of the Dtih point group as... 

This diabatization matrix only mixes the adiabatic states 2 and 4 leaving the states 1 and 3 unchanged. 

Comparing this equation with Eq. (75), it is seen that the mixing angle p is, up to an additive constant, identical to the relevant adiabatic-to-diabatic transformation—angle y ... 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

As explained in the Introduction, most mixed quantum-classical (MQC) methods are based on the classical-path approximation, which describes the reaction of the quantum degrees of freedom (DoF) to the dynamics of the classical DoF [9-22]. To discuss the classical-path approximation, let us first consider a diabatic... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

Abstract A mixed molecular orbital and valence bond (MOVE) method has been developed and applied to chemical reactions. In the MOVE method, a diabatic or valence bond (VE) state is defined with a block-localized wave function (ELW). Consequently, the adiabatic state can be described by the superposition of a set of critical adiabatic states. Test cases indicate the method is a viable alternative to the empirical valence bond (EVE) approach for defining solvent reaction coordinate in the combined qnantum mechanical and molecnlar mechanical (QM/MM) simulations employing exphcit molecular orbital methods. 

---