Lowdin matrix

The Lowdin matrix is constructed using the diagonal elements of the overlap matrix ... 

Lowdin now makes a detour to discuss the method of Cl. He expands the wave function in the set of Slater determinants that can be formed from the one-electron basis, j/. He introduces the word complete Cl to emphasize that the expansion is, in principle, exact. We have since learned to use the expression full Cl in the case where the one-electron basis is finite (Lowdin made a sarcastic comment about our use of the word complete in the complete active space (CAS) concept at a Sanibel Conference in 1980). He computes the matrix elements between the Slater determinants without assuming orthogonality between the MOs. These are the famous Lowdin matrix elements, which generalized the Slater formulae to the nonorthogonal case. 

A certain answer may be found (Lowdin 1955) by considering the first-order density matrix y(x x1) defined by Eq. II.9. By means of the basis y>k and formula III. 14, this matrix may be expressed in the form... 

We have presented above the derivation of eqns 38 and 39 in great detail because it includes expressions of general utility, in particular the variation of the eigenvectors (eqns 7 and 24) of an MO problem after Lowdin orthogonalization and the resulting variation of the population matrix P. The generalization to a Hamiltonian more complicated than that of eqn 19 is possible by following step by step the above derivation. 

The source matrices (for the sequence Fe C 0) are presented in Table 1 together with the resnlting derivative of the Lowdin popnlation matrix P. The electronegativities derived from the complete expression 38 and from the approximate expression 39 are -7.1690 eV and -7.3582 eV, respectively, thns snggesting that even in the unfavourable... 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

The inverse Lowdin transform constructed for the above idempotent matrix S(AT)i/2P(c AT), K)S(K) n- given with respect to the actual new, macromolecular overlap matrix S(K1), is expressed as... 

The chromophore atoms are assigned charges given by the density matrix in the Lowdin basis from the INDO/S calculation ... 

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... 

There is far less reported experience for ab initio studies of electronically excited states than for ground states. Matrix Hartree-Fock calculations for excited states cannot be considered routine. Often the same basis set is used for both the ground and excited state even though as long ago as 1958 Shull and Lowdin [17] demonstrated... 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

Thus the Fock matrix in the X basis is diagonalized by standard methods to yield the MO energies e and the matrix V from which the coefficient matrix c may be obtained by c = uV. There are several ways in which the matrix u and its inverse may be determined. The most commonly used is the symmetric othogonalization due to Lowdin, which involves diagonalization of the overlap matrix. We will not discuss this further. 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

Partitioning technique refers to the division of data into isolated sections and it was put into successful practice in connection with matrix operations. Lowdin, in his pioneering studies, [21, 22] developed standard finite dimensional formulas into general operator transformations, including treatments appropriate for both the bound state and the continuous part of the spectrum, see also details in later appendices. Complementary generalizations to resonance-type problems were initiated in Ref. [23], and simple variational formulations were demonstrated in Refs. [24,25]. Note that analogous forms were derived for the Liouville equation [26, 27] and further developed in connection with a retarded-advanced subdynamics formulation [28]. 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

---