Dipole expectation value

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

The dipole moment for a calculation is reported and is contained in the log file if logging is turn ed on. It is also reported on the status line if you Display DipoleMoment on iheDisplay menu. Other expectation values besides the dipole moment (for example, qna-dnipole moment ) could he reported with a wave function but the set reported with this release of HyperChem is limited to on ly a few. Below we discuss the properties or oth er ch aracteri/ation s of the calculated wave function that can he interactively visnali/ed. 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

In the absence of an external field, the unperturbed dipole and quadrupole moments may be calculated from the electronic wave function as simple expectation values. 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

When r is not large compared with the length of either dipole the value will be somewhat greater. For two molecules in water we may substitute r = 2.9 X 10-8 centimeter, and for /j insert the value from Table 41. Since the observed heat of evaporation of water is in the neighborhood of 0.5 electron-volt, we expect to find a value of the order of 0.25 electron-volt, since we are discussing here a molecule that has only one neighbor. We obtain... 

Given equilibrium quantum expectation values, we can calculate moments of the infra-red vibrational lineshape using a procedure originally outlined by Gordon.The infrared vibrational lineshape is given as the Fourier transform of the dipole moment correlation function ... 

The properties of a quantum mechanical system such as an AIM are readily calculated from any method as long as they involve an operator acting on the electron density, e.g., for the case of the dipole moment. The problem would seem to become harder for other properties, although the introduction of property densities allows us to generally introduce AIM expectation values [45], The expectation value of a property A for atom a in the Hirshfeld and QCT methods can be written as... 

Expectation Values for Zero-Field Energies, Virial Coefficients (r ), Dipole Moments (p), and Static Polarizabilites (a) for Non-BO LiH/D for Various Expansion Lengths (m) ... 

In order to obtain computational expressions for the polarizabilities, the time-dependent moments are evaluated as time-dependent expectation values of the dipole (ii and quadrupole operators y... 

In Table 1 the predicted dipole and quadmpole polarizability tensor components ay and C,y for the vibrational states with quantum number v are given. They were calculated for all vibrational states supported by the potential energy function as expectation values of the polarizability radial functions a(R) and C(R) over the vibrational wavefunction (equation (14)). The latter were obtained from... 

The integrations over the electronic coordinates contained in < >, as well as the integrations over vibrational degrees of freedom yield "expectation values" of the electric dipole moment operator because the electronic and vibrational components of i and [Pg.287]

In order to relate the dressed state population dynamics to the more intuitive semiclassical picture of a laser-driven charge oscillation, we analyze the induced dipole moment n) t) and the interaction energy V)(0 of the dipole in the external field. To this end, we insert the solution of the TDSE (6.27) into the expansion of the wavefunction Eq. (6.24) and determine the time evolution of the charge density distribution p r, t) = -e r, f)P in space. Erom the density we calculate the expectation value of the dipole operator... 

III.B. In order to analyze the wavefunction in a chemically more intuitive way, it is useful to localize it. In the framework of AIMD this is, for example, done by calculating the maximally localized Wannier functions (MLWF) and the corresponding expectation values of the position operator for a MLWF basis the so-called maximally localized Wannier centers (MLWCs), see Fig. 1 (67-72). With the help of the MLWC it is possible to compute molecular dipole moments (72-82). Furthermore, it is possible with the MLWC to obtain molecular properties, e.g., IR spectra (75,76,82-85). 

Notice that in equation (A.80) the dipole moment operator and the distances r,- and Ri are vectors which are usually expressed in Cartesian coordinates. The molecular dipole moment within the BO approximation is evaluated as an expectation value [recall equation (A.8)],... 

Molecular electronic dipole moments, pi, and dipole polarizabilities, a, are important in determining the energy, geometry, and intermolecular forces of molecules, and are often related to biological activity. Classically, the pKa electric dipole moment pic can be expressed as a sum of discrete charges multiplied by the position vector r from the origin to the ith charge. Quantum mechanically, the permanent electric dipole moment of a molecule in electronic state Wei is defined simply as an expectation value ... 

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