Density Integral Transform

Density Integral Transforms (DIT) and Molecular Quantum Simflarity Measures (MQSM)... 

Let us define an n-th order Density Integral Transform as the transformation of a density matrix element [4], of the same order ... 

P0 Every Quantum System in a given state can be described by a Density Integral Transform. It has... 

Until the advent of density functional theory (Chapter 13), thinking centred around means of circumventing the two-electron integral transformation, or at least partially circumventing it. The Mpller-Plesset method is one of immense historical importance, and you might like to read the original paper. 

The cumulants [2,43] of decay time sen are much more useful for our purpose to construct the probability P(t. xq)—that is, the integral transformation of the introduced probability density of decay time wT(t,xo) (5.2). Unlike the representation via moments, the Fourier transformation of the probability density (5.2)—the characteristic function—decomposed into the set of cumulants may be inversely transformed into the probability density. 

This equation is a partial differential equation whose order depends on the exact form of/ and F. Its solution is usually not straightforward and integral transform methods (Laplace or Fourier) are necessary. The method of separation of variables rarely works. Nevertheless, useful information of practical geological importance is apparent in the form taken by this equation. The only density distributions that are time independent must obey... 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

Litegrating over A between tlie limits 0 and 1 corresponds to smoothly transforming the non-interacting reference to the real system. This integration is done under the assumption that the density remains constant. 

The wave function W(x, i) may be represented as a Fourier integral, as shown in equation (2.7), with its Fourier transform A p, t) given by equation (2.8). The transform A p, i) is uniquely determined by F(x, t) and the wave function F(x, t) is uniquely determined by A p, i). Thus, knowledge of one of these functions is equivalent to knowledge of the other. Since the wave function F(x, /) completely describes the physical system that it represents, its Fourier transform A(p, t) also possesses that property. Either function may serve as a complete description of the state of the system. As a consequence, we may interpret the quantity A p, f)p as the probability density for the momentum at... 

Integration of the phase density over classical phase space corresponds to finding the trace of the density matrix in quantum mechanics. Transition to a new basis is achieved by unitary transformation... 

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