Schrodinger equation matrix representation

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 a first discretization step, we apply a suitable spatial discretization to Schrodinger s equation, e.g., based on pseudospectral collocation [15] or finite element schemes. Prom now on, we consider tjj, T, V and H as denoting the corresponding vector and matrix representations, respectively. The total... 

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. 

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] ... 

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... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrodinger equation is then written... 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

Properties of the 2-RDM and the V-Representability Ih-oblem The Matrix Contracting Mapping The Contracted Schrodinger Equation... 

As mentioned in Section I, Cho [13], Cohen and Frishberg [14, 15], and Nakatsuji [16] integrated the Schrodinger equation and obtained an equation that they called the density equation. This equation was at the time also studied by Schlosser [44] for the 1-TRDM. In 1986 Valdemoro [17] applied a contracting mapping to the matrix representation of the Schrodinger equation and obtained the contracted Schrodinger equation (CSE). In 1986, at the Coleman Symposium where the CSE was first reported, Lowdin asked whether there was a connection between the CSE and the Nakatsuji s density equation. It came out that both... 

This equation is the matrix representation of the Schrodinger equation in the N-electron space. In order to contract it into the two-electron space, we will apply the MCM to both sides of the equation and get... 

D. A. Mazziotti, Pursuit of A-representability for the contracted Schrodinger equation through density-matrix reconstruction. Phys. Rev. A 60, 3618 (1999). 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

While early work [16, 19] on the CSE assumed that Nakatsuji s theorem [37], proved in 1976 for the integrodifferential form of the CSE, remains valid for the second-quantized CSE, the author presented the first formal proof in 1998 [20]. Nakatsuji s theorem is the following if we assume that the density matrices are pure A-representable, then the CSE may be satisfied by and if and only if the preimage density matrix D satisfies the Schrodinger equation (SE). The above derivation clearly proves that the SE imphes the CSE. We only need to prove that the CSE implies the SE. The SE equation can be satisfied if and only if... 

When contracting the matrix form of the Schrodinger equation into the two-electron space and transforming into normal form the resulting equation, one obtains the 2-CSE [45, 50]. In the spin-orbital representation, the 2-CSE splits... 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

Let us emphasize that we have made no approximations yet. Equation (3.13) is a set of simultaneous differential equations for the coefficients cm that determine the state function (3.13) is fully equivalent to the time-dependent Schrodinger equation. [The column vector c(/) whose elements are the coefficients ck in (3.8) is the state vector in the representation that uses the tyj s as basis functions. Thus (3.13) is a matrix formulation of the time-dependent Schrodinger equation and can be written as the matrix equation ihdc/dt = Gc, where dc/dt has elements dcmf dt and G is the square matrix with elements exp(.iu>mkt)H mk. ... 

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