Matrix elements projected basis

Computational effort for computing matrix elements with symmetry-projected basis functions can be reduced by a factor equal to the order of the group by exploiting commutation of the symmetry projectors with the Hamiltonian and identity operators. In general. 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

It is worth mentioning that constant term disappears in Eq. (6) because of the suitable choice of the basis in the form (4). In the following, the evolution of the density matrix of the relevant system will be examined by means of the standard projection technique. The matrix elements pa/it) of the reduced density matrix operator are defined as follows... 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

The AIM electron-population displacements, d/V, are strongly coupled through the olf-diagonal hardness matrix elements //y y>,i Thus, a given displacement d/Vk strongly affects the chemical potentials of all AIM. This representation considers all AIM populational parameters as independent variables, which can be interpreted as projections of the populational vector (/V, d, + N2 2 + dm) onto the orthogonal system of populational axes associated with the constituent atoms, i.e., the AIM populational basis vectors ... 

Projection onto a limited basis, or subspace, entails a concommitant replacement of the conventional hamiltonian with an effective operator, some parts of which are necessarily energy dependent and also all other operators for the system must be replaced by effective operators chosen so that their matrix elements in the projected basis are exact. The first stage of our treatment - the primitive parameterization - explicitly neglects... 

The expressions for the matrix elements obtained in the preceding section, together with Eqs. (80)-(83), enable us to write implicit equations determining the cluster coefficients and the correlated energy in terms of the cluster coefficients and the one- and two-electron integrals over the spin-orbital basis. We may write Eq. (80), the projection of the Schro-dinger equation for the CCSDT wave function on the singly excited space, as... 

The curvature coupling elements are thus simply off-diagonal matrix elements of the unprojected force constant matrix in the basis of eigenvectors of the projected force constant matrix. The classical notion that a trajectory will overshoot the path and climb the wall if the path curves on the way down the hill is a reflection of this curvature coupling. Climbing the wall in a transverse direction is tantamount to exchanging energy between the reaction path and the transverse vibration. 

Although the analysis in terms of the propagators for independent motion gL is convenient for displaying the content of the kinetic theory expression for the rate kernel, calculations based on (10.4), which contains the propagator for the correlated motion of the AB pair, are probably more convenient to carry out. In kinetic theory, such rate kernel expressions are usually evaluated by projections onto basis functions in velocity space. (We carry out such a calculation in Section X.B). Hence the problem reduces to calculation of matrix elements of (coupled AB motion in a nonreactive system) and subsequent summation of the series. This emphasizes the point that a knowledge of the correlated motion of a pair of molecules for short distance and time scales is crucial for an understanding of the dynamic processes that contribute to the rate kernel. 

For an orthonormal basis set, all of the matrix elements Sgr equal zero. The basis defines a subspace of the total Hilbert space and a projection operator can be introduced through the kernel... 

To solve the BSE for finite systems, all the equations are projected onto an orthonormal spin-orbital basis ( )p. As the equations are four-point equations relating two-particle quantities, they are in fact projected onto the basis of products of two spin orbitals. Each matrix element is thus indexed by two double indices. 

As we did for hydrogen, we here ignore spin. Thus is the quantum number of rotation of the atoms about their center of mass, m is the projection of this angular momentum in the lab frame, and is its projection in the body frame. In this basis, the matrix elements of the Stark interaction are computed using Equation 2.10 to yield... 

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