The Molecular Hamiltonian Operator

The Hamiltonian operator H(r, R) of a molecule composed of AT nuclei and Ngi electrons is the sum of a nuclear kinetic energy operator (KEO) T (R), an electronic KEO Tgiir) and a potential energy operator V(r, R) that describes the Coulombic interaction between the different particles. Here, r and R denote vectors collecting all the electron and nuclear coordinates respectively. The KEOs read 

Quantum Dynamics and Laser Control for Photochemistry, 

The quantum mechanical description of a molecule requires the solution of the time-dependent Schrodinger equation (TDSE) associated with the Hamiltonian operator i/(r, R) = T u (R) + Tel (f) + y(f, R) described above 

Because the motions of the various particles are correlated through the potential terms of Eq. (2.3), the direct integration of the molecular Schrodinger equation is an extremely difficult task that is only possible, in practice, for the simplest atomic and molecular systems. 

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In the clamped-nucleus Born-Oppenheimer approximation, with neglect of relativistic effects, the molecular Hamiltonian operator in atomic units takes the form... 

There are two sides to the deperturbation process the Hamiltonian model and the partially assigned and analyzed spectrum. The molecular Hamiltonian operator must be organized in a way that, of the infinite number of molecular bound states and continua, emphasis is placed on the finite subset of levels sampled in the spectrum under analysis. A computational model is constructed in which the unknown information about the relevant states is represented by a set of adjustable parameters that may be systematically varied, until an acceptable match between calculated and observed properties is obtained. 

For the case of the Onsager model (spherical cavity, dipole moment only) the term added to the molecular Hamiltonian operator is given by eq. (14.65). 

Quantum mechanics describes system behavior in terms of operators that represent measurable quantities, their eigenfunctions, which describe possible states of the system, and their eigenvalues, which correspond to allowable values of the measurable. If the system is presumed best described in terms of a specification of the system energy, then one looks for states [f) that are eigenfunctions of the molecular Hamiltonian operator H. That is, one solves the problem... 

We now begin the stndy of molecular qnantum mechanics. If we assnme the nuclei and electrons to be point masses and neglect spin-orbit and other relativistic interactions (Sections 11.6 and 11.7), then the molecular Hamiltonian operator is... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

By definition, a synnnetry operation R connnutes with the molecular Hamiltonian //and so we can write the operator equation ... 

This definition causes the wavefiinction to move with the molecule as shown for the X direction in figure Al.4,3. The set of all translation synnnetry operations / constitiites a group which we call the translational group G. Because of the imifomhty of space, G is a synnnetry group of the molecular Hamiltonian //in that all its elements commute with // ... 

Her and Plesset proposed an alternative way to tackle the problem of electron correlation tiler and Plesset 1934], Their method is based upon Rayleigh-Schrddinger perturbation 3ty, in which the true Hamiltonian operator is expressed as the sum of a zeroth-er Hamiltonian (for which a set of molecular orbitals can be obtained) and a turbation, "V ... 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

The problem still is to finding a complete set of solutions to the molecular Hamiltonian H that contains nuclear kinetic energy operators ... 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

Of course, this is an approximation, and (6.105) is not an exact eigenfunction of the molecular Hamiltonian //, but is an eigenfunction of some operator which we shall call H°. (We shall not specify H° explicitly.) A better approximation to the true molecular wave function is obtained by applying perturbation theory, taking the perturbation Hamiltonian H as... 

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

The initial and final asymptotic states are always expanded in the time independent basis associated with the molecular hamiltonian the scattering matrix is unitary. Note again that the basis contains all possible resonance and compound states. If there is no interaction, the scattering matrix is the unit matrix 1. Formally, one can write this matrix as S= 1+iT where T is an operator describing the non-zero scattering events including chemical reactions. Thus, for a system prepared in the initial state Op, the probability amplitude to get the system in the... 

Moreover, using the eigenvalue equation (A.l) of the molecular Hamiltonian, and then suppressing the closeness relation between the dipole moment operator, the SD is transformed into... 

The kinetic energy of the mode Q being invariant, we may write for the molecular hamiltonian, using the excitation boson operators B, B ... 

Thus we see that the operator g is not strictly an angular momentum operator in the quantum mechanical sense, which is why we have assigned it a different symbol. More importantly for the present purposes, we cannot use the armoury of angular momentum theory and spherical tensor methods to construct representations of the molecular Hamiltonian. In addition, the rotational kinetic energy operator, equation (7.89), takes a more complicated form than it has for a nonlinear molecule where there are three Euler angles (rotational coordinates). 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

A chemical molecule, by contrast consists of many particles. In the most general case N independent constituent electrons and nuclei generate a molecular Hamiltonian as the sum over N kinetic energy operators. The common wave function encodes all information pertaining to the system. In order to constitute a molecule in any but a formal sense it is necessary for the set of particles to stay confined to a common region of space-time. The effect is the same as on the single confined particle. Their behaviour becomes more structured and interactions between individual particles occur. Each interaction generates a Coulombic term in the molecular Hamiltonian. The effect of these terms are the same as of potential barriers and wells that modify the boundary conditions. The wave function stays the same, only some specific solutions become disallowed by the boundary conditions imposed by the environment. 

In this paper, both theories will be briefly reviewed presenting their differences. From these comparisons, the Non-rigid Molecule Group (NRG) will be stricktly defined as the complete set of the molecular conversion operations which commute with a given Hamiltonian operator [21]. The operations of such a set may be written either in terms of permutations and permutations-inversions, just as in the Longuet-Higgins formalism, or either in terms of physical operations just as in the formalism of Altmann. But, the order, the structure, the symmetry properties of the group will depend exclusively on the Hamiltonian operator considered. 

The complete set of the molecular conversion operations which commute with such an Hamiltonian operator (18) will contain overall rotation operations describing the molecule rotating as a whole, and internal motion operations describing molecular moieties moving with respect to the rest of the molecule. Such a group is called the full Non-Rigid Molecule Group (full NRG). 

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