Expansion amplitude

It can be proven [31] that all possible Slater determinants of N particles constructed from a complete system of orthonormalized spin-orbitals 4>k form a complete basis in the space of normalized antisymmetric (satisfying the Pauli principle) functions, of N electrons i.e. for any antisymmetric and normalizable (K one can find expansion amplitudes so that ... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

The expansion amplitudes An in (1.4.3) have to be determined such that at time t = 0 the wave function fulfils the starting condition ... 

In the Anderson picture the suppression of classical chaotic diffusion is understood as a destructive phase interference phenomenon that limits the spread of the rotor wave function over the available angular momentum space. The localization effect has no classical analogue. It is purely quantum mechanical in origin. The localization of the quantum rotor wave function in the angular momentum space can be demonstrated readily by plotting the absolute squares of the time averaged expansion amplitudes... 

Inserting this expansion into the time dependent Schrodinger equation with the Hamiltonian (5.4.13) yields the following set of coupled equations for the expansion amplitudes A t)... 

With this definition the system of coupled equations for the expansion amplitudes reads ... 

The boundary conditions satisfied by the expansion amplitudes 7 (x) are given by... 

At 248 nm excitation, expansion and contraction dynamics of polyimide film is given in Figure 6 at the fluence of 20 mJ/cm. The irradiated film began to expand during the excimer laser pulse and contracted rapidly, ose rate was about 1 nm/ns and 0.1 nm/ns, respectively. The maximum expansion amplitude was attained when the excitation pulse ends, and the expansion disappeared completely after a few ms. 

Expansion dynamics at 351 nm excitation with the fluence of 80 mJ/cm and 130 mJ/cm below the ablation threstold is shown in Figure 7. The irradiated film began to expand during the excimer laser pulse similarly as at 248 nm excitation, and expansion amplitude increased with the fluence. The contraction took place quite slowly, except small bump at 130 mJ/cm, and then the original flat surface was recovered. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

It can be shown [ ] that the expansion of the exponential operators truncates exactly at the fourth power in T. As a result, the exact CC equations are quartic equations for the t y, etc amplitudes. The matrix elements... 

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

At this stage, we do not know how much of 2p.j, 3p.j, 4p.j,. .. np.j this wavefunction contains. To probe this question another subsequent measurement of the energy (corresponding to the H operator) could be made. Doing so would allow the amplitudes in the expansion of the above f"= P.i... 

Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflec ted, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through sohds in a complex array of types of elastic waves. Sound is charac terized by its amplitude, frequency, phase, and direction of propagation. 

If the combustion process within a gas explosion is relatively slow, then expansion is slow, and the blast consists of a low-amplitude pressure wave that is characterized by a gradual increase in gas-dynamic-state variables (Figure 3.7a). If, on the other hand, combustion is rapid, the blast is characterized by a sudden increase in the gas-dynamic-state variables a shock (Figure 3.7b). The shape of a blast wave changes during propagation because the propagation mechanism is nonlinear. Initial pressure waves tend to steepen to shock waves in the far field, and wave durations tend to increase. 

The thermostatic expansion valve is substantially an undamped proportional control and hunts continuously, although the amplitude of this swing can be limited by correct selection and installation, and if the valve always works within its design range of mass flow. Difficulties arise when compressors are run at reduced load and the refrigerant mass flow falls below the valve design range. It is helpful... 

A linear expansion of this equation for a small-amplitude potential modulation, SU, leads to the microwave reflectivity change... 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Inserting the perturbation and Fourier expansion of the cluster amplitudes and the Lagrangian multipliers,... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

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