Schrodinger equation matrices

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... 

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

By using this expression for G, it is possible to write the nuclear Schrodinger equation (A.8) in matrix form [54,179] as... 

Substitution of Eq. (12) into the Schrodinger equation leads to a system of coupled differential equations similai to Eq. (5), but with the following differences the potential matrix with elements... 

Although Eq. (139) looks like a Schrodinger equation that contains a vector potential x, it cannot be interpreted as such because t is an antisymmetric matrix (thus, having diagonal terms that are equal to zero). This inconvenience can be repaired by employing the following unitary bansformation ... 

The technique for this calculation involves two steps. The first step computes the Hamiltonian or energy matrix. The elements of this matrix are integrals involving the atomic orbitals and terms obtained from the Schrodinger equation. The most important con-... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

The interest of contracting the matrix form of the Schrodinger equation by employing the MCM, is that the resulting equation is easy to handle since only matrix operations are involved in it. Thus, when the MCM is employed up to the two electron space, the geminal representation of the CSchE has the form [35] ... 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

The results of this test of the TDB-FMS method are encouraging, and we expect the gain in efficiency to be more significant for larger molecules and/or longer time evolutions. Furthermore, as noted briefly before, the approximate evaluation of matrix elements of the Hamiltonian may be improved if we can further exploit the temporal nonlocality of the Schrodinger equation. 

Born-Huang expansion, 286—289 first-derivative coupling matrix, 290—291 nuclear motion Schrodinger equation, 289-290... 

By substituting the expression for the matrix elements in Eq. (B.21), we get the final form of the Schrodinger equation within the diabatic representation... 

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