Hamiltonian, average value

Using this notation, the Hamiltonian H of the system can be linked in an elegant way to a matrix H. Let us apply this form of the wave function for the calculation of the Hamiltonian average value. To simplify the expressions, let us limit ourselves to the truncated expansion ... 

This is not the Hamiltonian average value expression. Additionally, the operator inside the bracket is not hermitian. But, until we assume that (O Eq. 3.192) is true, such an approach works. We can also construct an integral ... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

The zero-field spin Hamiltonian parameters, D and E, are assumed to be distributed according to a normal distribution with standard deviations oD and aE, which we will express as a percentage of the average values (D) and (E). -Strain itself is not expected to be of significance, because the shape of high-spin spectra in the weak-field limit is dominated by the zero-field interaction. 

With this constraint we can expand the average value E of the energy of (9) over the Hamiltonian (10) ... 

If i and j are different nuclei (for instance 13C and H) the hetero-nuclear dipolar interaction, which is often strong, can be removed by dipolar decoupling, which consists of irradiation nucleus j (say H) at its resonance frequency while observing nucleus i (say 13C). The time-averaged value of the Hamiltonian is then zero. 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

The average value of the Hamiltonian Hln] with respect to an MCSCF or MRCI wave function may be written as... 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

The idea in the method of adiabatic elimination is that the time evolution of the components in // 1 oscillates very rapidly with respect to the evolution of the components in. This justifies the substitution of the time-dependent components in M1 by some average values. This leads then to an effective Hamiltonian in 0 that takes the form [see Ref. 39, p. 1166, Eq. (18.7-7), where there is a sign misprint]... 

This potential is that, adapted to a Morse potential in q, of the Hamiltonian defined in eqs. (5.5) and (5.A38) in the case of a harmonic potential in q. The Morse potential in q that enters eq. (6.2) is represented in Figure 6.1, together with the harmonic potential that has at its minimum at go = 1A the same curvature, defined by the parameter >. We should remember that these potentials are represented for a fixed value of = Qq. The effect of this ID anharmonicity (parameter 8) is to shift the average value of q towards higher values, increase its amplitude of vibration, and also shift all 0 n transitions towards lower wavenumbers... 

The internal spin interaction Hamiltonian Hmt can be decomposed into spatial Tm[ ua(l) ] and spin Sm degrees of freedom Hin (t) = 2mTm[ ua(t) ]Sm. The spatial contribution, hereafter an NMR interaction rank-2 tensor T, is a stochastic function of time Tm[ ua(t) because it depends on generalized coordinates < ( ) of the system (atomic and molecular positions, electronic or ionic charge density, etc.) that are themselves stochastic variables. To clarify the role of these coordinates in the NMR features, a simple model is developed below.19,20 At least one physical quantity should distinguish the parent and the descendant phase after a phase transition. For simplicity, we suppose that the components of the interaction tensor only depend on one scalar variable u(t) whose averaged value is modified from m to m + ( at a phase transition. To take into account the time fluctuations, this variable is written as the sum of three terms, i.e. u(t) — m I I 8us(t). The last term is a stationary stochastic process such that — 0, where <.) denotes a... 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

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