Matrix element transition gradient

The A matrix involves elements between singly excited states while B is given by matrix elements between doubly excited states and the reference. The P/Q elements are matrix elements of the operator between the reference and a singly excited state. If P = r this is a transition moment, and in the general case it is often denoted a property gradient , in analogy with the case where the operator is the Hamiltonian (eq. (3.67). 

The formulas above give the gradient and the Hessian in terms of matrix elements of the excitation operators. They can be evaluated in terms of one-and two-electron integrals, and first and second order reduced density matrices, by inserting the Hamiltonian (3 24) into equations (4 9), (4 11), and (4 13)-(4 15). Note that transition density matrices and are needed for the evaluation of the Cl coupling matrix (4 15). 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

In the quantum theory of absorption that we discuss in Chap. 5, the transition gradient matrix element V arises directly, rather than just as an... 

Figure 4.17 illustrates the functions that enter into the transition gradient matrix element for the HOMO—>LUMO excitation of ethylene. Contour plots of the HOMO and LUMO (V a and tpi,) are reproduced from Fig. 4.6B in panels A and B... 

Fig. 4.17 Components of the transition matrix element of the gradient operator for excitation of ethylene. (A, B) Contour plots of the amplitudes of the HOMO and LUMO molecular orbitals in the yz plane. The C=C double bond lies on the y axis. (C, E) The derivatives of with respect to y and z, respectively. (D, F) The products of these derivatives with The y and...

Fig. 4.18 Canonical orientations of orbitals of two carbon atoms. The shaded regions represent boundary surfaces of the wavefunctions as in Fig. 2.5. The Vy z Cartesian coordinate system is centered on atom 1 with the y axis aligned along the interatomic vector. The transition gradient matrix elements VVn and V y are for the orientations shown in A, B and C, respectively. Matrix elements for an arbitrary orientation can be expressed as linear combinations of these canonical matrix elements...

Fig. 9.4 Vector diagrams of contributions to the transition-gradient matrix elements for the first two excitations of trans-butadiene. The initial and final molecular wavefunctions in the four-orbital model used for Fig. 4.19 are indicated. The atoms are labeled 1 by the empty circles...

In the above, /(r) = (c/87r) o( ) the intensity of the laser beam at the point r Is = cl4n) h yld) is the saturation intensity d=d- e is the projection of the dipole moment matrix element of the polarization vector e of the laser beam A is the detuning of the laser field frequency cu with respect to the atomic transition frequency tuo, that is, A = lo — loo, and the quantity 27 defines the rate of spontaneous decay of the atom from the upper level e) to the lower level g), that is, the Einstein coefficient A. Figure 5.6 shows the dependence of the radiation pressure force and the gradient force on the projection v =v of the atomic velocity on the propagation direction of a Gaussian laser beam for the case of strong saturation of the D-line of Na. 

The IRC step in a flexible subset of coordinates away from the transition state also follows from subdividing the step vector and the Jacobian matrix. Begin with the IRC equation in mass-weighted Cartesian coordinates, Eq. (7). Substitute for using Eq. (66), but note that only a subset of J is required, since = 0 for each stiff coordinate A. The elements of J that multiply a nonzero value of htf correspond exactly to the elements of Jj defined in Eq. (70). Similarly, only the gradients with respect to flexible generalized coordinates are nonzero... 

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