CSF method

Surface Tension Simulation in Diffuse Interface Method Similar to surface tension modeling in the VOF method, both CSF and CSS methods are applicable, with minor adjustments, to the diffuse interface method. The CSF method is as follows ... 

Once the surface between the fluids has been determined, there remain two important additional steps before surface forces can be included. First, the curvature of the surface must be calculated from the discrete location values available on the computational grid. Then the surface tension force is applied by replacing a surface force by an equivalent volume force in a zone adjacent to the interface, with most commercial software using a version of the Continuum Surface Force (CSF) method derived by Brackbfll et al. [40]. The manner in which this force is implemented can lead to the appearance of spurious velocities, known as parasitic currents. A recent study by Harvie et ol. [41] has shown that these parasitic currents cannot be removed by mesh refinement alone and they can have a significant effect in certain conditions. 

Qian and Lawal [87[ used the VOF model in Fluent 6.1 (a basic VOF model with the CSF method for the surface tension force see Section 5.3.1.2) to study Taylor bubble formation via a T-junction made from circular ducts with diameters in the... 

According to the applied Continuum Surface Force (CSF) method we approximate the Dirac Delta function 5 by a continuous regularized one, which is a smooth function in the vicinity of e of the interface ... 

The third term on the left hand side of (18.1) is an artificial compression term which is active only in the interface region. U ei is a velocity field suitable to compress the interface. The gas-liquid flow is governed by the incompressible Navier—Stokes (N-S) equations in which the parameters about physical properties such as density ip) and dynamic viscosity (//) are calculated as weighted averages by the linear interpolation of the volume fraction. The continuum surface force (CSF) method is employed to calculate the surface tension force [2]. Therefore, the N-S equatimis can be expressed as follows ... 

If polymer samples with a certain desirable molecular weight are not directly available, like with some biopolymers, fractionation is often the only way to achieve such produrts. Eckelt and Wolfr have recently given an overview concerning the fractionation of biopolymers by means of CSF. This chapter describes the direct fractionation of native cellulose in the solvent-nonsolvent system DMAc+LiCl/acetone. How the CSF method can be applied for the polysaccharides pulluian and dextran is outlined by Eckelt et al. ... 

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

So-called complete active space (CAS) methods fomi all CSFs that... 

Methods that are based on making the fiinctional (T // T ) / ( T T ) stationary yield upper bounds to the lowest energy state having the synnnetry of the CSFs in T. The Cl and MCSCF methods are of this type. 

D) MOST PERTURBATION AND CC METHODS ARE SIZE-EXTENSIVE, BUT DO NOT PROVIDE UPPER BOUNDS AND THEY ASSUME THAT ONE CSF DOMINATES... 

As larger atomic basis sets are employed, the size of the CSF list used to treat a dynamic correlation increases rapidly. For example, many of the above methods use singly- and doubly-excited CSFs for this purpose. For large basis sets, the number of such CSFs (N ) scales as the number of electrons squared uptimes the number... 

MCSCF methods describe a wave function by the linear combination of M configuration state functions (CSFs), with Cl coefficients, Ck,... 

The above expansion of the full N-eleetron wavefunetion is termed a "eonfiguration-interaetion" (Cl) expansion. It is, in prineiple, a mathematieally rigorous approaeh to expressing F beeause the set of aU determinants that ean be formed from a eomplete set of spin-orbitals ean be shown to be eomplete. In praetiee, one is limited to the number of orbitals that ean be used and in the number of CSFs that ean be ineluded in the Cl expansion. Nevertheless, the Cl expansion method forms the basis of the most eommonly used teehniques in quantum ehemistry. 

The configuration interaction (CI) method in whieh the LCAO-MO eoeffieients are determined first (and independently) via either a single-eonfiguration SCF ealeulation or an MCSCF ealeulation using a small number of CSFs. The CI eoeffieients are subsequently determined by making the expeetation value < F H F >/< F I F >... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

So-ealled complete-active-space (CAS) methods form all CSFs that ean be ereated by distributing N valenee eleetrons among P valenee orbitals. For example, the eight non-eore eleetrons of H2O might be distributed, in a manner that gives Ms = 0, among six... 

In the CI method, one usually attempts to realize a high-level treatment of electron correlation. A set of orthonormal molecular orbitals are first obtained from an SCF or MCSCF calculation (usually involving a small to moderate list of CSFs). The FCAO-MO... 

The implementation of the CC method begins mueh as in the MPPT/MBPT ease one seleets a referenee CSF that is used in the SCF proeess to generate a set of spin-orbitals to be used in the subsequent eorrelated ealeulation. The set of working equations of the CC teehnique given above in Chapter 19.1.4 ean be written explieitly by introdueing the form of the so-ealled eluster operator T,... 

The CC method, as presented here, suffers from the same drawbaeks as the MPPT/MBPT approaeh its energy is not an upper bound and it may not be able to aeeurately deseribe waveflinetions whieh have two or more CSFs with approximately equal amplitude. Moreover, solution of the non-linear CC equations may be diffieult and slowly (if at all) eonvergent. It has the same advantages as the MPPT/MBPT method its energy is... 

Photoelectron spectra have confirmed the expected trends in the frontier orbitals.The tetrafiuoro derivative 12.12 (R = F) is prepared by treatment of C6F5SNSNSiMc3 with CsF in acetonitrile (Scheme 12.2). Several difiuoro- and trifiuoro-benzodithiadiazines have also been prepared by these methods.In contrast to 12.12 (R = H), which has an essentially planar structure in the solid state,the dithiadiazine ring in the tetrafiuoro derivative is somewhat twisted. In the gas phase, on the other hand, electron diffraction studies show that 12.12 (R = F) is planar whereas 12.12 (R = H) is non-planar. ... 

As illustrated above, even quite small systems at the CISD level results in millions of CSFs. The variational problem is to extract one or possibly a few of the lowest eigenvalues and -veetors of a matrix the size of millions squared. This cannot be done by standard diagonalization methods where all the eigenvalues are found. There are, however, iterative methods for extraeting one, or a few, eigenvalues and -veetors of a large matrix. The Cl problem eq. (4.6) may be written as... 

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