Hamiltonian operator coordinate wave function

The true Hamiltonian operator and wave function involve the coordinates of all n electrons. The Hartree-Fock Hamiltonian operator E is a one-electron operator (i.e., it involves the coordinates of only one elec-... 

The true Hamiltonian operator and wave function involve the coordinates of all n electrons. The Hartree-Fock Hamiltonian operator F is a one-electron operator (that is, it involves the coordinates of only one electron), and (13.148) is a one-electron differential equation. This has been indicated in (13.148) by writing F and as functions of the coordinates of electron 1 of course, the coordinates of any electron could have been used. The operator F is peculiar in that it depends on its own eigenfunctions [see Eqs. (13.149) to (13.152)], which are not known initially. Hence the Hartree-Fock equations must be solved by an iterative process. 

Here He is the electronic Hamiltonian determining the wave functions and the eigenvalues of the Fe-(NO) fragment in the fixed nuclear configuration. Reading off the coordinate qA of the full symmetric vibration from its equilibrium position in the state 1Ai(Fe2+(d6)), one can write down the operator of the linear electron-vibrational interaction in the following form ... 

From the separability theorem (Section 1.6.3) it follows that if an operator (e.g. the Hamiltonian) depending on N coordinates can be written as a sum of operators that only depend on one coordinate, the corresponding N coordinate wave function can be written as a product of one-coordinate functions, and the total energy as a sum of energies. 

Of more general interest are the selection rules for S, L, and /, as these quantum numbers describe an atomic state with greater accuracy. To the approximation that we neglect spin in the Hamiltonian operator, the spin wave functions are independent of the coordinate wave functions. The dipole moment integral will vanish because of the orthogonality of the spin functions unless the spin quantum numbers match in the initial and final states. To this approximation, we thus have the selection rule AS == 0 that is, only transitions between terms of the same multiplicity are allowed. The selection rules for L and J cannot be derived so simply the results are ... 

A convenience of electronic basis functions (53) is that they reduce at infinitesimal-amplitude bending to (28) with the same meaning of the angle 9 we may employ these asymptotic forms in the computation of the matrix elements of the kinetic energy operator and in this way avoid the necessity of carrying out calculations of the derivatives of the electronic wave functions with respect to the nuclear coordinates. The electronic part of the Hamiltonian is represented in the basis (53) by... 

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

The power of quantum theory, as expressed in Eq. (4.1), is that if one has a molecular wave function in hand, one can calculate physical observables by application of the appropriate operator in a manner analogous to that shown for the Hamiltonian in Eq. (4.8). Regrettably, none of these equations offers us a prescription for obtaining the orthonormal set of molecular wave functions. Let us assume for the moment, however, that we can pick an arbitrary function,, which is indeed a function of the appropriate electronic and nuclear coordinates to be operated upon by the Hamiltonian. Since we defined the set of orthonormal wave functions 4, to be complete (and perhaps infinite), the function must be some linear combination of the 4>,, i.e.,... 

The complete, nonrelativistic Hamiltonian for a diatomic molecule is given by (1.272). If one inverts the Cartesian coordinates of all particles (nuclei and electrons), then H in (1.272) is unchanged, since all interparticle distances are unchanged. Thus the parity operator IT commutes with this Hamiltonian, and we can characterize the overall wave function of a diatomic molecule by its parity. (This statement applies to both homonuclear and heteronuclear diatomics.)... 

There is no operator of differentiation on R in Hamiltonian (3), and hence these coordinates are the parameters. The wave function i///i(r R) obeys the following Schrodinger equation... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Closely inspecting the operator terms entering the electronic Hamiltonian eq. (1.27) one can easily see that they are sums of equivalent contributions dependent on coordinates of one or two electrons only. Analogously in the second quantization formalism only the products of two and four Fermi operators appear in the Hamiltonian. Inserting the trial. Y-electron wave function of the (ground) state into the expression for the electronic energy yields its expectation value in terms of the expectation values of the one- and two-electron operators ... 

---