Energy independent wave operators

The Energy-Independent Wave Operators and the Non-Linear Form of the Schrodinger Equation. 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

The generalized Bloch equation (12) is the basis of the RS perturbation theory. This equation determines the wave operator and, together with Eq. (11), the energy corrections for all states of interest especially, it leads to perturbation expansions which are independent of the energy of the individual states, just referring the unperturbed basis states. Another form, better suitable for computations, is to cast this equation into a recursive form which connects the wave operators of two consecutive orders in the perturbation V. To obtain this form, let us start from the standard representation of the Bloch equation (16) in intermediate normalization and define... 

To obtain an explicit cancelation of the disconnected terms in the Eqs. 3.35 and 3.36 it is necessary to separate those terms out when they appear due the action of the Cl operator, C, on 0). We should note that the use of the semi-linear form of the CASCCSD wave function (3.32) leads to equations which are explicitly dependent on the energy the system, while the fully exponential form of the wave function, (3.12), leads to energy independent equations. We should also add that the CASCC wave function can be alternatively expressed in the following pure linear form ... 

Within the energy-independent Bloch formalism (Section II.A.3), the reduced wave operator (33) can be written as... 

Within the energy-independent Bloch approach, the reduced wave operator becomes... 

One way to address this problem is to treat the two configurations as a model space, and use the approach of Jeziorski and Monkhorst (1981) to determine the correlated wave function and energy. In this approach, the reference determinants are correlated independently with the unrestricted wave operator form, except that excitations internal to the open-shell space are excluded. Thus, the f operator is... 

In the above discussion we have been concerned with the exact electronic Hamiltonian, energies and wave functions of a supersystem consisting of an array of well-separated subsystems. We now turn our attention to the description afforded by some independent particle model, in which the electrons move in some mean field. The most commonly used approximation of this type is the Hartree-Fock model, but the discussion presented in this section is not restricted to this particular method. In particular, we write the total electronic Hamiltonian operator in the form... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

Here, d is the electric dipole operator, tp (,v) are the wave functions of the intragap states with energies w/2, and C is an -independent coefficient (for small w, we can neglect the weak tw-dependence of the real pan of the dielectric constant). 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

---