Symmetry of the Hamiltonian

Since the Laplacian V2 is invariant under orthogonal transformations of the coordinate system [i.e. under the 3D rotation-inversion group 0/(3)], the symmetry of the Hamiltonian is essentially governed by the symmetry of the potential function V. Thus, if V refers to an electron in a hydrogen atom H would be invariant under the group 0/(3) if it refers to an electron in a crystal, H would be invariant under the symmetry transformations of the space group of the crystal. 

Consider the operation P, that corresponds to some coordinate transformation T, on the Schrodinger equation 

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The individual values of the exponents are detennined by the symmetry of the Hamiltonian and the dimensionality of the system. 

For large this sum is again dominated by the first eigenvalue, Ai, which will now depend on M. For practical calculations M is restricted by computer memory. However, the symmetry of the Hamiltonian allows a block... 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

In HMO theory the assumed separability is partially justified by the symmetry of the Hamiltonian. In the present instance there is no such symmetry and neglect of interaction terms is tantamount to assuming a classical Hamiltonian. 

One distinguishes between three different types depending on space-time symmetry classified by the Dyson parameter (3d = 1,2,4 (Guhr, Muller-Groeling and Weidenmuller, 1998). The Gaussian orthogonal ensemble (GOE, (3d = 1) holds for time-reversal invariance and rotational symmetry of the Hamiltonian... 

The time-reversal symmetry of the Hamiltonian dynamics, also called the microreversibility, is the property that if the phase-space trajectory... 

The structure of the reduced density matrix follows from the symmetry properties of the Hamiltonian. However, for this case the concurrence C iJ) depends on ij and the location of the impurity and not only on the difference i—j as for the pure case. Using the operator expansion for the density matrix and the symmetries of the Hamiltonian leads to the general form... 

Since the nuclear coordinates appearing in Eq. 3.2 are fixed parameters, as indicated by using the upper-case symbol R for the intemuclear distances, the spatial symmetry of the Hamiltonian is reduced to those operations that leave the nuclear framework invariant. (Permutational symmetry among the electrons is retained and will be considered in Chapter 6.)... 

The group T SP] will be found important for the symmetry of the hamiltonian of the rotation-internal nuclear motion problem associated with SRMs (Sect. 3.2). 

Because of the symmetry of the Hamiltonian integrals involving ipa T) and 7>a (2) are identical and calculated only once. 

Of course, this is an artifact of the calculation because the probability of having an electron with a and /3 spin is the same. Kraka et al.55 demonstrated that the on-top density discussed by Perdew et al.56 is able to reproduce the correct symmetry of the Hamiltonian. 

Fukutome11 in a series of papers has made a thorough analysis of the symmetry properties of the GHF spin orbitals. As pointed out by Low-din,12 there is no reason why a solution D of (2.23) should be symmetry adapted to the symmetry of the Hamiltonian. We can impose symmetry constraints on the spin orbitals, but that will in general raise the total energy. To characterize this situation Lowdin has coined the term symmetry dilemma. ... 

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 symmetry properties with which we are concerned arise from the symmetry of the Hamiltonian in Hiickel theory, this means the symmetry of the effective, one-electron Hamiltonian ( 2.2). Clearly, this has the same symmetry as the molecular framework itself. 

In this section we show how the symmetry of the Hamiltonian can be used to simplify the relaxation equations. We also present several important theorems involving time-correlation functions and memory functions. We begin by discussing time reversal symmetry. 

A word of caution is necessary. In the presence of an external field the time-reversal symmetry of the Hamiltonian may be removed. For example, in a magnetic field B the spin-dipole interaction with the field is of the form(—J B), where J is an angular momentum and B is the magnetic field. J has odd time-reversal symmetry (like rxp) so that if B 0 none of the above theorems hold unless B —> — B is also imposed. Thus for example Theorem 3 would become... 

Some of the consequences of the spherical symmetry of the hamiltonian of an atomic system for the Green s function are explored in this section. A number of results concerning tensor operators are invoked and the reader are assumed to be familiar with the Wigner 3j and 6j symbols. 

We might compare the preceding treatment of H2 with the perturbation treatment of the helium ls2s levels (Section 9.7). There we started with the degenerate functions ls (l) (2) and ls(2)2s(l). Because of the symmetry of the Hamiltonian with respect to interchange of the identical electrons, we found the correct zeroth-order functions to be [ls (l)2s(2) ls(2)2s(l)]/V2. For H2,we started with the degenerate functions Is and Isj. Because of the symmetry of the electronic Hamiltonian with respect to the identical nuclei, we found the correct zeroth-order functions to be Is, ls,)/V2(l ... 

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