Hamiltonian operator definition

It is convenient to introduce the dimensionless variable by the definition so that the Hamiltonian operator becomes... 

To gain an understanding of this mechanism, consider the Hamiltonian operator (H — Egl) with only two-body interactions, where Eg is the lowest energy for an A -particle system with Hamiltonian H and the identity operator I. Because Eg is the lowest (or ground-state) energy, the Hamiltonian operator is positive semi-definite on the A -electron space that is, the expectation values of H with respect to all A -particle functions are nonnegative. Assume that the Hamiltonian may be expanded as a sum of operators G,G,... 

In the Kohn Sham equations (A.116) [324, 325], the core Hamiltonian operator h( 1) has the same definition as in HF theory (equation A.6), as does the Coulomb operator, 7(1), although the latter is usually expressed as... 

The coordinates are expressed in the molecular axes x,y,z, whidi are rigidly attached to the molecule. These coordinates and masses are labelled in some laboratory axes, X,Y,Z, fixed in space. Under this definition, a symmetry operation is a change of axes that leaves the Hamiltonian operator (14) invariant, and the group of all such operations is the Schrodinger subgroup. 

To clear up this definition let us consider a molecule in absence of any external perturbation, so its nuclear Hamiltonian operator can be written in terms of relative coordinates with respect to the center of mass, neglecting the translation coordinates. In this conditions, the nuclear Hamiltonian operator is written as ... 

It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation. 

Here we manipulated the left side of the inequality by using the corresponding definition and the Lagrange parameter associated with the Hamiltonian operator H. For the sake of clarity, we have here exhibited the parameter dependence of the appropriate ensemble density operators along with those associated with the respective minimum free energies and entropies corresponding to the two Hamiltonians. 

In accordance with the classical definition of the Hamiltonian, eqn (8.52), and recalling that the derivative of the Lagrangian with respect to a velocity (eqn (8.51)) is the momentum conjugate to the corresponding coordinate, the Hamiltonian operator is defined as... 

A special word on the calculation of the energy is in order. In the last equality in (If) the hamiltonian operator in the numerator can be pushed either to the left or to the right in such a way to operate directly on the trial state. Remembering the definition of the local energy, (4), we can write... 

A unique feature of the occupation number representation is that the number of electrons does not appear in the definition of the Hamiltonian operator in this form as it does in the wavefunction form. This is because all of the occupation information resides in the bras and kets. This is true for any operator in second quantized form. This feature is used to advantage in theories that allow the number of particles to change, and to a more limited extent in the calculation of electron affinities and ionization potentials. It is less important to the MCSCF method but it is useful to remember that the bras and kets contain all of the occupation information. Other details of the wavefunction, such as the AO and MO basis set information, are included in the integrals that are used as expansion coefficients in the second quantized representation of the operator. 

The second form, the normal order of the generator product, shows that the operator also preserves and eigenvalues since it is constructed from operators that do so. The expansion of the Hamiltonian operator in this spinpreserving operator basis shows that the Hamiltonian operator itself must preserve the and eigenvalues of the wavefunctions on which it acts. The definition of the operator results in the identities... 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Previous work has not investigated if commutation relations are conserved upon transformation to effective operators. Many important consequences emerge from particular commutation relations, for example, the equivalence between the dipole length and dipole velocity forms for transition moments follows from the commutation relation between the position and Hamiltonian operators. Hence, it is of interest to determine if these consequences also apply to effective operators. In particular, commutation relations involving constants of the motion are of central importance since these operators are associated with fundamental symmetries of the system. Effective operator definitions are especially useful... 

Effective Hamiltonians and Effective Operator Definitions Corresponding to a Time-Independent Operator A... 

Mappings Norms Conserved" Effective Hamiltonian Effective operator definition Effective Hamiltonian is the A = H Case State- Independent... 

We now turn to the effective operator definitions produced by (2.14) with model eigenfunctions that incorporate the normalization factors of (2.16) so their true counterparts are unity normed. Equations (2.27) and (2.38) show these model eigenfunctions to be the a)o and ( that are defined in (2.33) and (2.34). Substituting Eqs. (2.27) and (2.38) into (2.14) and proceeding as in the derivation of the forms / = I-I1I, yields the state-independent definitions A, A" and A" of Table I. Notice that the effective Hamiltonian H is identically produced upon taking A = // in the effective operator A". Table I indicates that this convenient property is not shared by all the effective operator definitions. 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

The commutation relation between two arbitrary operators is not conserved upon transformation to effective operators by any of the definitions. Many state-independent effective operator definitions preserve the commutation relations involving // or a constant of the motion, as well as those involving operators which are related to P in a special way, for example, A with [P, 4] = 0. Many state-dependent definitions also conserve these special commutation relations. However, state-dependent definitions are not as convenient for formal and possibly computational reasons. The most important preserved commutation relations are those involving observables, since, as discussed in Section VII, they ensure that the basic symmetries of the system are conserved in effective Hamiltonian calculations. 

Semi-empirical Hamiltonians and operators are taken to be state independent [56] and have the same Hermiticity as their true counterparts. Consequently, the valence shell effective Hamiltonians and operators they mimic must also have these two properties. Table I shows that the effective Hani iltonian and operator definitions H and A, as well as H and either A or a fulfill these criteria. Thus, these definition pairs may be used to derive the valence shell effective Hamiltonians and operators mimicked by the semi-empirical methods. Table III indicates that the commutation relation (4.12) is preserved by all three definition pairs. Hence, the validity of the relations derived from the semi-empirical version of (4.12) depends on the extent to which the semi-empirical Hamiltonians and operators actually mimic, respectively, exact valence shell effective Hamiltonians and operators. In particular, the latter Hamiltonians and operators contain higher-body terms which are neglected, or ignored, in semi-empirical theories. These nonclassical higher body interactions have been shown to be nonnegligible for the valence shell Hamiltonians of many atoms and molecules [27, 145-149] and for the dipole moment operators of some small molecules [56-58]. There is no a... 

The mathematical issues relevant to the definition of density functional derivatives can be considered in the simple model of noninteracting electrons. As in the KSC [4], this singles out the kinetic energy. The /V-electron Hamiltonian operator is H = T + V. Orbital functional derivatives determine the noninteracting OEL equations... 

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