Partitioned hamiltonian matrix

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

The theory of the Bk method [22] is based on the partitioning technique in perturbation theory [23, 24]. Suppose the Hamiltonian matrix H of the MR-CI space is partitioned as... 

In these equations it is also assumed that the orthogonal complement states have been chosen to diagonalize the Hamiltonian matrix. This is merely a formal convenience and is not necessary for the validity of the results of this section. Substitution of these identities into the expression for the partitioned orbital Hessian matrix gives... 

Consider the Hiickel Hamiltonian-matrix, HI, for an alternant hydrocarbon, constructed on the basis of the simple Huckel-approximations (equations (6-2)-(6-5)) as an example, the matrix, HI, for butadiene is shown in equation (2-54). If the (m) starred atoms are labelled from 1, 2,..., rn, and the unstarred ones from m + 1, m + 2,. .., n, then by an exactly similar argument to that used when discussing the corresponding secular-determinant for an alternant hydrocarbon in 6.3, the matrix HI may be partitioned as in equation (D7). 

This equation expresses the fact that the extended Green s function Q u>) is the projection of an operator resolvent onto a set of )U-orthonormal states [10]. Note that the matrix is hermitian if the Hamiltonian H of the many-body system is hermitian (which is assumed throughout this paper). By matrix partitioning we can write for the inverse of the Green s function... 

Iss), (21si — ls2 — Iss), (ls2 — Iss). (We continue to ignore overlap between Is AOs.) Since they know that it is not possible for bases of different symmetries to mix (it would produce an MO of mixed symmetry), they know at once that they can have a MOs from mixtures of N2s, A2p i d (1/V3)(lsi + ls2 + Iss) and -type MOs from the remaining functions. In effect, they have used symmetry to partition their functions into two subsets that do not interact with each other. This means that, in the MO calculation, the hamiltonian matrix will have no mixing elements between members of different subsets. This is indicated schematically in Fig. 13-16. They also know in advance that there will be three MOs of a symmetry (since only three basis functions have that symmetry) mid four of e symmetry (two degenerate pairs). It is interesting to see how strongly symmetry controls the nature of NH3 MOs. 

After dividing the determinants into subsets defined by their Mg values, we order the subsets from highest to lowest Mg value. The subsets define a partitioning of the Hamiltonian matrix H into blocks. Determinants from sets whose Mg values differ by more than two have zero Hamiltonian matrix elements between them, because the excitation between them is more than a two-electron excitation, and the Hamiltonian contains at most two-electron operators. With the arrangement of the determinants in Mg blocks from highest to lowest, H is therefore block pentadiagonal. This structure is shown in figures 10.1 and 10.2 for an even and an odd number of electrons. 

For an optimized coupled-cluster state, we may thus write the EOM-CC Hamiltonian matrix in the partitioned form... 

Hamiltonian matrix H represents the operator (P -I- Q)H(P -I- Q) (i.e. the projection of the exact H on the fuU-Cl space. Partition the operator and its eigenfunctions, writing the eigenvalue equation in 2-component form, and eliminate the component QIP as in Section 2.5.]... 

This is the version of X2C as it has been implemented in NWChem [14], Due to the RKB condition, for simplicity the code currently requires a fiilly uncontracted basis set. It has been demonstrated that local decoupling schemes are suitable whereby an atomic and nearest-neighbor partitioning is employed in order to render the constmction of the X2C Hamiltonian matrix less CPU and memory intensive [53,85,86]. 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

It remains to compute the matrix elements < Xi H Xj > To do that we partition the Hamiltonian into the zeroth order plus the first order contribution. This partitioning is state dependent ... 

Here a and b are occupied MO s of systems A and B. Equation (6,32) is easily expressible in terms of integrals over atomic basis functions and elements of the density matrix. In eqn. (5.31) two terms may be distinguished. The first one is due to single electron excitations of the type a r") and (b —->s"), where a and r", respectively, are occupied and virtual MO s in the system A, and b and s" are occupied and virtual MO s in the system B, Contribution of these terms corresponds to the classical polarization interaction energy, Ep, Two-electron excitations (a r", b — s"), i.e. simultaneous single excitations of either subsystem, may be taken as contributions to the second term - the classical London dispersion energy, Ep, If the Mjiller-Ples-set partitioning of the Hamiltonian is used, Ep may be expressed in... 

The differential cross section for ionisation is given by (6.60). To formulate the T-matrix element we partition the total Hamiltonian H into a channel Hamiltonian K and a short-range potential V and use the distorted-wave representation (6.77). The three-body model is defined as follows. 

Equations [76] only allow for isotropic fluctuations in the volume. However, it is sometimes useful to allow the lengths and angles of the simulation box all to vary separately. Fully flexible cell simulations of this type, first carried out by Parrinello and Rahman, can also be formulated in rms of a non-Hamiltonian dynamical scheme. In such a scheme, the matrix h representing the cell, which contains the cell vectors in its columns, is incorporated as a dynamical variable. That is, nine extra variables are added to the phase space along with an additional nine from its corresponding momentum Pg matrix. In terms of the box matrix, the partition function A(N, P, T) is given by... 

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