Hamiltonian operator electrons

At this point, it is appropriate to make some conmrents on the construction of approximate wavefiinctions for the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is given by the general fonn... 

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and for each molecule defined in tenns of its assigned electrons. The unperturbed Hamiltonian for the system is then 0 = - A perturbation XH consists of tlie Coulomb interactions between the nuclei and... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

The total Hamiltonian operator H must commute with any pemiutations Px among identical particles (X) due to then indistinguishability. For example, for a system including three types of distinct identical particles (including electrons) like Li2 Li2 with a conformation, one must satisfy the following commutative laws ... 

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 reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

There are cases in which the angular momentum operators themselves appear in the Hamiltonian. For electrons moving around a single nucleus, the total kinetic energy operator T has the form ... 

We need to be clear about the various coordinates, and about the difference between the various vector and scalar quantities. The electron has position vector r from the centre of mass, and the length of the vector is r. The scalar distance between the electron and nucleus A is rp, and the scalar distance between the electron and nucleus B is tb- I will write / ab for the scalar distance between the two nuclei A and B. The position vector for nucleus A is Ra and the position vector for nucleus B is Rb. The wavefunction for the molecule as a whole will therefore depend on the vector quantities r, Ra and Rb-It is an easy step to write down the Hamiltonian operator for the problem... 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is... 

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

The one-electron Hamiltonian operator h = h +v, with kinetic energy operator, generates a complete spectrum of orbitals according to the Schrodinger equation... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

The helium atom serves as a simple example for the application of this construction. If the nucleus (for which Z = 2) is considered to be fixed in space, the Hamiltonian operator H for the two electrons is... 

The unperturbed Hamiltonian operator is the sum of two hydrogen-like Hamiltonian operators, one for each electron... 

Perturbation terms in the Hamiltonian operator up to still lead to the uncoupling of the nuclear and electronic motions, but change the form of the electronic potential energy funetion in the equation for the nuclear motion. Rather than present the details of the Bom-Oppenheimer perturbation expansion, we follow instead the equivalent, but more elegant procedure of M. Bom and K. Huang (1954). 

The nonrelativistic, electronic Schrodinger Hamiltonian operator, designated as H, is represented by... 

The paramagnetism of the triplet state can be observed by electron spin resonance spectroscopy. This is perhaps the most reliable means of determining the existence of a triplet state since the ESR signals can be predicted using the following Hamiltonian operator ... 

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