Total vector multiplication

Let us develop a performance model for the parallel matrix-vector multiplication. We first note that if the dimensions of A are n x n, the maximum number of processes that can be utilized in this parallel algorithm equals n. The total number of floating point operations required is rr (where we have counted a combined multiply and add as a single operation), and provided that the work is distributed evenly, which is a good approximation if n p, the computation time per process is... 

We will use this algorithm instead of the all-to-all broadcast algorithm provided in the employed implementation of MPl because the latter algorithm displayed very irregular performance. A performance model for a matrix-vector multiplication that uses the all-to-all broadcast is discussed in section 6.4.1. The total execution time, the speedup, and the efficiency can then be expresssed as the following functions of p and n... 

The FMM naturally fits the DDA, since the matrix-vector multiplication is actually computing the total field on each single dipole due to all other dipoles [79]. And it was actually implemented... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

Secondly, you must describe the electron spin state of the system to be calculated. Electrons with their individual spins of sj=l/2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin S=Esj. Since spin is a vector, there are various ways of combining individual spins, but the net result is that a molecule can have spin S of 0, 1/2, 1,. These states have a multiplicity of 2S-tl = 1, 2, 3,. ..,that is, there is only one way of orienting a spin of 0, two ways of orienting a spin of 1/2, three ways of orienting a spin of 1, and so on. 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Here the summation of charges times position vectors is replaced by the integral over the total wavefunction T (the square of the wavefunction is a measure of charge) of the dipole moment operator (the summation over all electrons of the product of an electronic charge and the position vectors of the electrons). To perform an ab initio calculation of the dipole moment of a molecule we want an expression for the moment in terms of the basis functions r/j, their coefficients c, and the geometry (for a molecule of specified charge and multiplicity these are the only variables in an ab initio calculation). The Hartree-Fock total wavefunction T is composed of those component orbitals i// which are occupied, assembled into a Slater determinant (Section 5.2.3.1), and the i// s are composed of basis functions and their coefficients (Sections 5.3). Equation (5.206), with the inclusion of the contribution of the nuclei to the dipole moment, leads to the dipole moment in Debyes as (ref. [lg], p. 41)... 

Solution of these equations leads naturally to the principal quantum number n and to two more quantum numbers, / and m The total energy of the electron is determined by n, and its orbital angular momentum by the azimuthal quantum number l. The value of the total angular momentum is /(/ + ) 2h. The angular momentum vector can be oriented in space in only certain allowed directions with respect to that of an applied magnetic field, such that the components along the field direction are multiples of fi the multiplying factors are the mi quantum... 

In the triplet state, the excited electron has the same spin orientation (parallel or antiparallel to the external field) as the electron in the original ground-state orbital, so that the state is paramagnetic with total spin S = 5, -I- S2 = 1. Its multiplicity (i.e. the number of quantum mechanically allowed projections of the spin vector 5 on the field B) is then 25 + 1 = 3, with magnetic quantum numbers = 0, 1. 

The form assumed by the continuity equation, equation (1-19), can be derived formally by integration over the total cross-sectional area of the flow. The limits of the coordinates X2 and X3 [which appear in equation (1-13)] in such an integration must be independent of x because the boundaries of the cross section are streamlines and must therefore be parallel to the local X coordinate (that is, parallel to the local velocity vector). Thus, since the flow variables are independent of the coordinates X2 and X3, multiplication of equation (1-19) by 2 3 followed by integration over the cross-sectional area shows that... 

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