Orthonormality, derivative Schrodinger equation

Orthonormality is a constraint that may be incorporated into the derivative Schrodinger equation or imposed separately [116]. 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

In order to evaluate 5s/ISv from Eq. (282), we further need the functional derivatives dqfjjdvs and ScpflSv. The stationary OPM eigenfunctions (< /r), = 1,..., oo) form a complete orthonormal t, and so do the time-evolved states unperturbed states, remembering that at t = ti the orbitals are held fixed with respect to variations in the total potential. We therefore start from t = ti, subject the system to an additional small perturbation (5i)s(r, t) and let it evolve backward in time. The corresponding perturbed wave functions [Pg.135]

It is important that the orbitals r ) that satisfy (5.17) are orthogonal for different eigenvalues We can use them to construct an orthonormal set with which to express the many-electron problem. We have assumed orthogonality in deriving (5.17). We now show that this is consistent by considering (5.17) as an equation for the matrix elements formed for the bra orbital ( from a Schrodinger equation for rj). The set of such equations for N different rj) is called the Hartree—Fock equations. 

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