Density matrix transition

When rh rfi, this expression defines an element of the p-order transition density matrix (p-TRDM) [2]. In what follows when T = T, one instead of two upper indices denoting th will be used. 

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 ... 

If we transform the MO s such that condition (5 11) is fulfilled, the resulting transition density matrix will be obtained in a mixed basis, and can subsequently be transformed to any preferred basis The generators Epq of course have to be redefined in terms of the bi-orthonormal basis, but this is a technical detail which we do not have to worry about as long as we understand the relation between (5 9) and the Slater rules. How can a transformation to a bi-orthonormal basis be carried out We assume that the two sets of MO s are expanded in the same AO basis set. We also assume that the two CASSCF wave functions have been obtained with the same number of inactive and active orbitals, that is, the same configurational space is used. Let us call the two matrices that transform the original non-orthonormal MO s [Pg.242]

To obtain the Hamiltonian matrix the matrix elements 1 Vsoiv. i > have to be computed. With the one-electron transition density matrix, T, also calculated by CASSI, this involves the following summation ... 

Then the same procedure as in the construction of the natural orbital basis for the HF calculations is used, and the truncated set of natural orbitals that is obtained is used to expand the transition density matrix, and hence the index-pair (k, /) is contracted. In practice this means that instead of making one large transformation with the transition density matrix in Eq. (9-19), two smaller ones are done first one from the ACBS to truncated orbital basis, then to state basis. For reasonable thresholds in the orbital construction, td, this will lead to a speed-up of the QMSTAT calculations. Details of the two procedures described above can be found in reference [139],... 

A linear transformation of a configuration vector cb thus requires the construction of a configuration gradient with B> as the reference state [Eq. (94)], and the construction of an orbital gradient with a symmetric transition density matrix [Eq. (95)]. A linear transformation on an orbital vector °b requires the construction of a configuration gradient [Eq. (96)] and an orbital gradient [Eq. (97)] from the one-index transformed Hamiltonian K. 

Density matrices of the state functions provide a compact graphical representation of important microscopic features for second order nonlinear optical processes. The transition moment y is expressed in terms of the transition density matrix p jji(r,r ) by nn " /j, ptr Pjj t(r,r )dr and the dipole moment difference Ay by the difference density function p - p between the excited and ground state functions = -e / r( p -p )dr where p is the first order reduced density matrix. 

Figure 5. Contour diagram of difference density functions for p - Pt (5a) and p - pjy (5b) and of transition density matrix Pl.IV (5c) ( ).

An advantage of using the non-orthogonal configuration basis is that the coupling coefficients are simple linear combinations of transition density matrix elements between internal N-electron states. These can in principle be... 

The (transition) density matrix structure of the coupling coefficients suggests that it should be possible to evaluate them in a similar way as proposed by Siegbahn " for CASSCF wavefunctions (cf. Section II.H) and to make use of factorizations such as... 

A feature of the template or index-driven approach, which was first proposed by Shavitt for direct Cl algorithms ", is that all contributions to a particular density matrix element, or may be computed together. In the MCSCF procedure, it is also useful to compute quantities of the form <0 erstul ) for a fixed (rstu) and for all possible n>. MCSCF procedures that do not use such a method must instead sort the list of coupling coefficients, which are computed in some arbitrary order, into an order that allows the orderly computation of these transition density matrix elements. The index-driven approach avoids this unnecessary sorting step in those cases where the coupling coefficients are explicitly written to an external file, and it allows the efficient computation of the required coefficients in those cases where they are used as they are computed. The relative merits of the index-driven and CSF-driven approaches are discussed further in Section VI. 

It is useful to compare these approaches when applied to a wavefunction expansion that results in a sparse density matrix. For example with a PPMC expansion, each d , with about possible unique elements, contains only about non-zero elements m(m -I- 2)/8 non-zero elements of the typ)e and mil non-zero elements of the type For m = 20 the matrix d" is only 0.29% non-zero. The inner product CSF-driven approach is clearly not suited for the sparse transition density matrix resulting from this type of wavefunction. The outer product CSF-driven approach does account for the density vector sparseness but the effective vector length is only n, the orbital basis dimension. 

The parameters NLW and NUW are the number of lower walks and upper walks for a particular Shavitt loop. (The mapping vector R( ) is loop-independent and gives the correspondence between the full set of upper and lower walks.) The walk offsets YB and YK are determined by the loop shape, and the coefficient T is a linear combination of the loop values and integrals associated with the Shavitt loop. Since only unique Shavitt loops are constructed, the innermost DO loop must be repeated with the bra and ket values interchanged, resulting in even more arithmetic operations for each Shavitt loop. The transition density matrix construction involves an analogous DO loop structure with the last statement replaced by... 

The single-particle transition density matrix connecting states K and L of molecule D is defined as usual [76] ... 

From the orbitals and slates obtained in question 4 new one- and two-electron integrals and one- and two-electron density and transition density matrix elements may now be evaluated, and the iterative procedure thus continued. The multiconfigurational HF total energies obtained during this iterative procedure are... 

Calculate the nonvanishing one- and two-electron density and transition density matrix elements of the form... 

It is clear that the matrix elements of the one- and two-body generators between Gel fand states and generate one- and two-particle transition density matrix elements respectively yf " and F with appropriate indices ... 

Therefore, we discuss the effect of LMCT in a different perspective. There are some difficulties for extracting LMCT effects from the Cl wave functions or the TDMs. Firstly, the evaluation of the weights of LMCT mixing in Cl wave functions is difficult because most virtual MOs contained in CSFs are delocalized between Ln and X. Secondly, the amount of mixing of LMCT CSFs is too small to analyze because the weight of the reference 4f CSFs exceeds 95 percent. Therefore, to remove the dominant 4f components, we focus on the Nocc x (Nact + Nya) rectangular block of transition density matrix elements, where Nocc, act. and T/vk are the numbers of the doubly occupied MOs, active MOs (4f), and virtual MOs, respectively. This rectangular block... 

In this appendix, we describe the detailed definition of y. The Nocc X ( act + JVvir) rectangular block transition density matrix T is translated to diagonalized rectangular matrix by using the corresponding orbital [77] or natural transition orbital [78, 79] method as follows. 

This parameter y of LnX3 takes a positive value because the amount of mixing of LMCT CSFs is much larger than that of MLCT CSFs. This weight of A is a diagonalized transition density matrix element and has the information of the products of the Cl coefficients, i.e., those for the... 

The antisymmetry of d (X) is a consequence of the orthonormality of the molecular orbitals, Eq. (43a). Here the adjective square has been emphasized in reference to the one-particle transition density matrix. The one particle transition density matrix is in general not symmetric, that is, the full or square matrix must be retained. However, in most electronic structure applications the associated one electron integrals, for example are symmetric, permitting the off-diagonal density matrix element to be stored in folded or triangular form. Since d is not symmetric, it is necessary to construct and store the transition density matrix in its unfolded or square form. 

In evaluating the CSF contribution to fc(X), fc(X), the fact that two orbitals have been differentiated must be considered. This gives rise to a contribution from the square two particle transition density matrix in addition to a contribution from the square one particle transition density matrix. In particular,... 

The transition density matrix (17) can be expanded in any convenient orbital basis as... 

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