Crude-adiabatic electronic

In this section we introduce the basic concept of diabatic electronic states and compare them with the adiabatic and crude adiabatic electronic wave-functions. 

To overcome the difficulty associated with this singularity, so-called crude-adiabatic electronic wavefunctions have been introduced into the literature.Here, wavefunctions Qo) = for a fixed nuclear configuration Qo are used in the expression (4) for any Q of interest ... 

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

Further we can proceed similarly as in the case of adiabatic approximation. We shall not present here the details, these are presented in [21,22]. We just mention the most important features of our transformation (46-50). Firstly, when passing from crude adiabatic to adiabatic approximation the force constant changed from second derivative of electron-nuclei interaction ufcF to second derivative of Hartree-Fock energy Therefore when performing transformation (46-50) we expect change offeree constant and therefore change of the vibrational part of Hamiltonian... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Diabatic electronic states (previously termed crude adiabatic states ) are defined as slowly varying functions of the nuclear geometry in the vicinity of the reference geometry [9-11]. The final vibronic-coupling Hamiltonian is obtained by adding the nuclear kinetic-energy operator which is assumed to be diagonal in the diabatic representation. 

In this representation, the molecular wavefunction is expanded using the electronic wavefunctions with the contiguration fixed at the reference configuration Rq. This representation is called a crude adiabatic (CA) representation and the basis Ro) the electronic basis. The other representation, the Born-Oppenheimer (BO) representation, is defined as... 

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

At a conical intersection, the branching plane is invariant through any unitary transformation within the two electronic states and any such combination of degenerate states is still a solution. Thus, the precise definition of the two vectors in (A.9) or (A. 11) is not unique and depends on an arbitrary rotation within the space of the Cl coefficients (i.e., between the generalized crude adiabatic states), unless the states have different symmetries (then xi is totally symmetric and X2 breaks the symmetry). 

Apart from special models, Eq. (14) is satisfied only when the whole space of interacting electronic states is considered in the ADT matrix S (in diatomics, with only a single nuclear degree of freedom, there is no curl-condition). This, however, is contradicting the spirit of choosing a small subset of electronic states in the ADT matrix and would lead one back to the crude adiabatic basis discussed in the previous section. Therefore strictly diabatic electronic states, satisfying rigorously Eqs. (12) and (13) do not exist in the multidimensional case. ... 

Formally, the off-diagonal appearance of W" in the diabatic basis is similar to the crude adiabatic basis [as is the (nearly) diagonal form of Tjv]. However, it should be kept in mind that the form of Eq. (18) refers to a small electronic subspace only, whereas all other electronic states are treated in the adiabatic representation. By contrast, the matrix W of Eq. (8) in the... 

In this equation Vtot(Q r) represents the entire potential energy, where we have already integrated over the electronic coordinates. The simplest of the various adiabatic representations is the "crude adiabatic" representation, in which the aromatic wave functions are found at a single value of f, which we will call o- One separates Vtot into two portions, V (Q) = Vtot(Q o) and AV(Q,f) = VtotCQJ) - V (Q), and then solves an r-independent Schrodinger equation for the aromatic wave function... 

The divergence of nonadiabatic couplings in Eq. 8.7 leads to computational problems in fuU quantum treatments, like those reviewed in this chapter, due to the difficulty to integrate them over the vibrational wavefunctions. Different strategies have been proposed to face these problems. Probably the most traditional solution is to work within the so-called crude-adiabatic approximation, where the electronic wavefunctions at a fixed nuclear configuration go are used in the expansion in Eq. 8.5,... 

Two main assumptions underlie the GF method (1) full separation of electronic and vibrational degrees of freedom (crude adiabatic approximation) and (2) local-... 

In the crude adiabatic approximation with fixed nuclei, the electronic Hamiltonian He does not depend on... 

Schematic drawing illustrating these aspects in case of NbsGe is presented in Fig. 27.6. The Fu phonon mode covers out-of phase stretching vibration of two perpendicular Nb chains in two planes - see Fig. 27. Id. For simplicity, drawing of only a single chain of Nb atoms in a plane (e.g. b-c plane) is sketched in Fig. 27.6. For equilibrium high-symmetry structure (Req) on the crude-adiabatic level, the highest electron density is localized at equilibrium position of Nb atoms in a chain - Fig. 27.6a. For distorted nuclear geometry (Rd,cr) in the Fn mode, electron density is polarized and the highest value is shifted into the inter-site positions-bipolarons are formed. The Fig. 27.6b corresponds to compression period in stretching vibration of Nbl-Nb2 which induces increase of Nbl-Nb2 inter-site electron density and decreases of Nb2-Nb3 electron density. For an expansion period. Fig. 27.6c, situation is opposite. Inter-site electron density is decreased for Nbl-Nb2 and increased for Nb2-Nb3. On the lattice scale, increase and decrease of electron density is periodic. On the adiabatic level, alternation of electron density is bound to vibrations at equilibrium nuclear positions (Fig. 27.6a-c).

In Fig. 27.7, iso-density lines of highest electron density calculated for Nb3Al on crude-adiabatic level by computer code SOLID2000 [55] are shown. 

In the crude adiabatic (CA) approximation [1,32-40], the electronic wavefimctions

specific nuclear configuration qg satisfy the following Schrodin-ger equation ... 

The electronic wavefunction in the crude adiabatic approximation is defined according to Equation 1.18 at a specific nuclear configuration q and therefore it does not... 

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