Electronic Hamiltonian symmetry operators with

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Because symmetry operators commute with the electronic Hamiltonian, the wavefunctions that are eigenstates of H can be labeled by the symmetry of the point group of the molecule (i.e., those operators that leave H invariant). It is for this reason that one constructs symmetry-adapted atomic basis orbitals to use in forming molecular orbitals. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-electron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Ep sp P commute with total Lz =Ej Lz (i), as in the linear-molecule case. The additional symmetry present in the spherical atom reflects itself in the fact that Lx, and Ly now also commute with H and A. However, since Lz does not commute with Lx or Ly, new quantum... 

Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "block diagonal". That is, objects of different symmetry will not interact only interactions among those of the same symmetry need be considered. 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

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 inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

According to Fig. III.8, ethylene oxide contains two mirror planes perpendicular to one another, the a, -plane m-ab) and the 6,c-plane (mb and the line of intersection of the two mirror planes generates a twofold axis, Czb- Due to this symmetry of the nuclear Coulomb potential, the electronic Hamiltonian, Eq. (IV.55a), is symmetric with respect to the operations ... 

A symmetry operator leaves the potential energy unchanged if it moves each electron so that after the motion it is the same distance from each nucleus or the same distance from another nucleus of the same charge as it was prior to the motion. If this is the case, the symmetry operator commutes with the Bom-Oppenheimer Hamiltonian. There is another way to see if the symmetry operator will commute with the electronic Hamiltonian. Apply it to the nuclei and not to the electrons. If the symmetry operator either leaves a nucleus in the same position or places it in the original position of a nucleus of the same type, it belongs to the nuclear framework. The symmetry operators that belong to the nuclear framework will commute with the electronic Hamiltonian when applied to the electrons. 

Since these functions are symmetry-adapted approximations to the eigenfunctions, we need only operate on any one with the two-electron Hamiltonian and integrate to return the energy of the corresponding triplet or singlet state (47, 77, 78), which result then can be optimized in terms of the parameters in the linear combinations. 

All of the above hinges on the assumption that /i is a representation of an operator h which is invariant with respect to the operations of the molecular point group. Obviously the one-electron Hamiltonian and the unit operator satisfy this condition and so the matrices h and S of the LCAO method can be symmetry blocked in this way. We have seen that, in general, the matrix representation of the Hartree-Fock operator will not satisfy this condition, so that it is a constraint on the LCAO method to make the assumption that the matrix can be treated in this way. As we have seen, this constraint consists of generating self-consistent symmetries which in certain critical cases may prevent us from obtaining the lowest-energy determinant. 

The Hiickel model as applied to polyenes possesses a symmetry known as alternancy symmetry, since the polyene system can be subdivided into two sublattices such that the Hiickel resonance integral involves sites on different sublattices. In such systems, the Hamiltonian remains invariant when the creation and annihilation operators at each site are interchanged with a phase of +1 for sites on one sublattice and a phase of -1 on sites of the other. Even in interacting models this symmetry exists when the system is half-filled. The alternancy symmetry is known variously as electron-hole symmetry or charge-conjugation symmetry [16]. 

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