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Functions basis

As Chapter 3 describes, the basis functions for each irreducible representation are limited in number. Basis functions are polynomial functions with specific behaviour under symmetry operations. Thus, in Ih, the set x, y, z transforms as components of the Tm representation. It is useful to have explicit basis functions to display the properties of the representations. [Pg.22]

The permutation characters (F, ) on orbits other than the regular orbit decompose into fewer irreducible components, for example, F, of the 60 vertices of Ceo is the direct sum [Pg.24]

The utility of such lists is emphasised in a further example, the construction of basis functions transforming with Tiu symmetry for the two orbit fullerene, Cso. The 80-vertex structure is formed by combination of the O20 orbit and an Oeo orbit of In point symmetry to realize the three-valent fullerene. The permutation character over the full set of 80 vertices is the direct sum [Pg.24]

Over the 80 vertices, taken as a whole, all three sets of the basis functions in Table 1.1 would be required, but it is simpler to carry out a two-orbit analysis, taking two copies of ITiu, the first over 60 vertices and the second over 20, with the third Tiu set being the projections of 2Tiu on the vertices of Oeo- [Pg.24]

In operation of the GT Calculator, a shorthand notation is used to display polynomial functions for all of the basis functions required by the regular representations of the point groups. This notation was suggested by Elert. The polynomial Cmnpx y zP is written Cmnp(mnp), [Pg.24]

In practice, numerical solution of the KS differential equation (71) typically proceeds by expanding the KS orbitals in a suitable set of basis functions and solving the resulting secular equation for the coefficients in this expansion and/or for the eigenvalues for which it has a solution. The construction of suitable basis functions is a major enterprise within electronic-structure theory (with relevance far beyond DFT), and the following lines do little more than explaining some acronyms often used in this held. [Pg.39]

The situation is quite similar in chemistry. Due to decades of experience with Hartree-Fock and Cl calculations much is known about the construction of basis functions that are suitable for molecules. Almost all of this continues to hold in DFT — a fact that has greatly contributed to the recent popularity of DFT in chemistry. Chemical basis functions are classified with respect to their behaviour as a function of the radial coordinate into Slater type orbitals (STOs), which decay exponentially far from the origin, and Gaussian type orbitals (GTOs), which have a gaussian behaviour. STOs more closely resemble the true behaviour of atomic wave functions [in particular the cusp condition of Eq. (19)], but GTOs are easier to handle numerically because the product of two GTOs located at different atoms is another GTO located in between, whereas the product of two STOs is not an STO. The so-called contracted basis functions , in which STO basis functions are reexpanded in [Pg.39]

A very popular approach to larger systems in DFT, in particular solids, is based on the concept of a pseudopotential (PP). The idea behind the PP is that chemical binding in molecules and solids is dominated by the outer (valence) electrons of each atom. The inner (core) electrons retain, to a good approximation, an atomic-like configuration, and their orbitals do not change much if the atom is put in a different environment. Hence, it is possible to approximately account for the core electrons in a solid or a large molecule by means of an atomic calculation, leaving only the valence density to be determined self-consistently for the system of interest. [Pg.40]

43Note that the effective potential vs is a way to deal with the electron-electron interaction. The pseudopotential is a way to deal with the density of the core electrons. Both potentials can be profitably used together, but are conceptually different. [Pg.40]

44This external PP is also called the unscreened PP, and the subtraction of //[ ] and from vfp[n t is called the unscreening of the atomic PP . It can only be done exactly for the Hartree term, because the contributions of valence and core densities are not additive in the xc potential (which is a nonlinear functional of the total density). [Pg.40]

The r (T)v in eq. (3) are the elements of the /th column of the matrix representative F(7) of the symmetry operator T. A realization of eq. (3) in 3-D space was achieved when the matrix representative (MR) of R((j z) was calculated in Section 3.2. The MRs form a group representation, which is either an irreducible representation (IR) or a direct sum of IRs. Let d sf be a set of degenerate eigenfunctions of // that corresponds to a particular eigenvalue E, so that [Pg.96]

The valence orbitals taken for a molecular-orbital calculation of a transition metal complex are the nd, (n + l)s, and (n + l)p metal orbitals and appropriate a and n functions of the ligands. Many of these valence orbitals are not individually basis functions for an irreducible representation in the symmetry under consideration. Symmetry basis functions transforming properly must be constructed, by methods analogous to those used throughout this volume. The results for a number of important symmetries are tabulated in this volume in various places, as follows  [Pg.107]

The basis functions referred to in Section 8-7 are normalized assuming zero ligand-ligand overlap. In reality, of course, the ligand valence orbitals overlap, and this should be taken into consideration when normalizing the basis functions. As an example, consider one of the T2 (ag) functions for a tetrahedral molecule (see Table 8-6). For as of T2, row 1  [Pg.107]

C4V Table I, Ballhausen-Gray reprint D4h Table II, Gray-Ballhausen reprint [Pg.107]

The decomposition of D ) into representations of the group O is now obtained directly from Eq. (42) with the aid of the character table (Table 3). [Pg.30]

The parity of D may be deduced from the fact that the d-orbitals, which are basis functions for D contain the spherical harmonics which do not change sign upon reflection in the origin. Therefore the representation D ) is of even parity. This property is carried over into the cubic group so that we may now say that D has been reduced to the representations eg and t2g in the group Oh, where the suffix g indicates even parity. [Pg.31]

It is now necessary to construct linear combinations of the d-orbitals which transform according to the representations eg and t2g. Generally speaking, the construction of basis functions may be quite tedious, apart from a number of simple cases where it may be done practically by inspection (as for example, the one-dimensional representations). Basis sets for the common situations are tabulated in various places e. g., Koster et al. (34), Ballhausen (2), Griffith (21). It will be sufficient for our purpose to give the basis sets for eg and t2g and to demonstrate that they satisfy the necessary requirements. Since our discussion is confined to systems of d-electrons, all states will be of even parity or g-states. To simplify the notation we shall henceforth suppress the parity index, unless specifically needed. [Pg.31]

We shall illustrate the behavior of the basis functions under the operations of the group 0. For convenience take the z-axis along one of the 4-fold symmetry axes as shown in Fig. 17. The six operations C4 are CJ, CJ, Q. For illustrative purposes we take Q. This operation may be described by [Pg.32]

We note the important fact that under Q there is no mixing of e-orbitals with t2-orbitals. The above relations may then be expressed in matrix form [Pg.33]


Nair, S., S. Udpa, and L. Udpa, (1993), Radial basis functions network for defect sizing . Review of Progress in QNDE, Vol. 12, 1993, pp. 819-825... [Pg.104]

The linear integral equation (5) is solved by a standard technique, including expansion of the unknown An z) by some basis functions and transformation of (5) into a matrix equation to... [Pg.128]

Caleulations that employ the linear variational prineiple ean be viewed as those that obtain the exaet solution to an approximate problem. The problem is approximate beeause the basis neeessarily ehosen for praetieal ealeulations is not suffieiently flexible to deseribe the exaet states of the quantnm-meehanieal system. Nevertheless, within this finite basis, the problem is indeed solved exaetly the variational prineiple provides a reeipe to obtain the best possible solution in the space spanned by the basis functions. In this seetion, a somewhat different approaeh is taken for obtaining approximate solutions to the Selirodinger equation. [Pg.46]

We now show what happens if we set up tire Hamiltonian matrix using basis functions i ), tiiat are eigenfiinctions of Fand with eigenvalues given by ( equation A1.4.5) and (equation Al.4.6). We denote this particular choice of basis fiinctions as ij/" y. From (equation Al.4.3). (equation A1.4.5) and the fact that F is a Hemiitian operator, we derive... [Pg.139]

The Hamiltonian matrix factorizes into blocks for basis functions having connnon values of F and rrip. This reduces the numerical work involved in diagonalizing the matrix. [Pg.139]

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. [Pg.140]

Although the Sclirodinger equation associated witii the A + BC reactive collision has the same fonn as for the nonreactive scattering problem that we considered previously, it cannot he. solved by the coupled-channel expansion used then, as the reagent vibrational basis functions caimot directly describe the product region (for an expansion in a finite number of tenns). So instead we need to use alternative schemes of which there are many. [Pg.975]

A double-zeta (DZ) basis in which twice as many STOs or CGTOs are used as there are core and valence AOs. The use of more basis functions is motivated by a desire to provide additional variational flexibility so the LCAO-MO process can generate MOs of variable difhiseness as the local electronegativity of the atom varies. [Pg.2171]

Dupuis M, Rys J and King H F 1976 Evaluation of molecular integrals over Gaussian basis functions J. Chem. Phys. 65 111-16... [Pg.2195]

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... [Pg.2195]

Bacic Z, Kress J D, Parker G A and Pack R T 1990 Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. 4. discrete variable representation (DVR) basis functions and the analysis of accurate results for F + Hg d. Chem. Phys. 92 2344... [Pg.2324]

Longuet-Higgins [7] also reinforces the discussion by tbe following qualitative demonstration of a cyclic sign change for the LiNaK like system subject to Eq. (3), in which rows and columns are labeled by the basis functions... [Pg.11]

The Hermite basis functions (p, t) have the following form ... [Pg.59]

In order to incorporate the geometric phase effect in a formulation based on an expansion in G-H basis functions we need to consider the operation of the momentum operator on a basis function, that is, to evaluate terms as... [Pg.76]

In the basis set formulation, we need to evaluate matrix elements over the G-H basis functions. We can avoid this by introducing a discrete variable representation method. We can obtain the DVR expressions by expanding the time-dependent amplitudes a (t) in the following manner ... [Pg.77]

Extension to six dimensions is now straightforward. We obtain similar expressions just with the y and z components and the index n running over the basis functions included in the particular degree of freedom. For the functions... [Pg.77]

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. [Pg.228]

In a completely general and flexible application of END one may choose to include some number, say Njon, of nuclei described as in Eq. (20) completely void of electronic basis functions, and some number N/ ) of nuclei with elechonic basis functions, as well as some number Np) of free centers. [Pg.230]

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. [Pg.267]

The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... [Pg.295]

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... [Pg.297]

Further, the time-independent electionic basis functions are taken to be the eigenfunctions of the electionic Hamiltonian,... [Pg.312]

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... [Pg.411]

This type of basis functions is frequently used in popular quantum chemishy packages. We shall discuss the way to evaluate different kinds of matrix elements in this basis set that are often used in quantum chemistt calculation. [Pg.411]

Next, we consider the simple overlap integral of two such basis functions with different powers of Cartesian coordinates and different Gaussian width, centered at different points. Let nuclei 1 locate at the origin, and let nuclei 2 locate at —R, then... [Pg.412]

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... [Pg.413]

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... [Pg.437]


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Artificial neurons Basis functions

Atom-Centered Basis Function Approach

Atomic basis function

Atomic orbital basis function

Atomic orbital basis function derivatives

Atomic orbital basis function incomplete

Averages Basis” thermodynamic functions

B-spline basis functions

Basis Functions Gauss-Type

Basis Functions Slater-Type

Basis Sets for Electronic Wave Functions

Basis cartesian functions

Basis expansion Slater-type functions

Basis function Slater

Basis function atom-centred functions

Basis function contracted

Basis function cubic harmonic

Basis function desireable properties

Basis function for atoms

Basis function generating machine

Basis function global

Basis function monomial

Basis function orthonormal

Basis function practical criteria

Basis function radial

Basis function spatial

Basis function, mathematical descriptions

Basis functions Cartesian Gaussian

Basis functions Coulomb potential derivatives

Basis functions Gaussian lobe

Basis functions Pople

Basis functions Schrodinger equation

Basis functions algebraic approach

Basis functions constructing transfer state function

Basis functions correlation consistent

Basis functions energy

Basis functions exchange correlation

Basis functions field dependent

Basis functions first-order derivatives

Basis functions floating

Basis functions formalism

Basis functions function Slater-type orbitals

Basis functions general considerations

Basis functions interaction terms between

Basis functions large-component

Basis functions linear models

Basis functions matrix elements

Basis functions node-based

Basis functions normalization factor

Basis functions numerical orbitals

Basis functions overlap integrals

Basis functions perturbative handling

Basis functions scalar

Basis functions second-order derivatives

Basis functions semiempirical methods

Basis functions tetraatomic molecules

Basis functions triatomic molecules

Basis functions, choice

Basis functions, construction

Basis functions, energy-dependent

Basis functions, interaction site fluids

Basis functions, representations generated

Basis polarization functions

Basis set diffuse functions

Basis set for correlated wave functions

Basis set polarization functions

Basis sets Gaussian functions

Basis sets Slater-type functions

Basis sets and the canonical ensemble partition function

Basis sets incorporating diffuse functions

Basis sets midbond functions

Basis sets/functions

Basis sets/functions computational quantum mechanics

Basis sets/functions superposition error

Basis spherical harmonic functions

Basis-Set Expansions of Relativistic Electronic Wave Functions

Bloch basis functions

Bond index basis functions

Cartesian Gaussian functions, in basis set

Cartesian Hermite Gaussian basis functions

Construction of basis functions

Correlated basis functions for large

Correlated basis functions for large molecules

Coulomb basis functions

Density functional theory basis Slater determinants

Density functional theory fundamental basis

Density-functional theory basis sets

Derivative Continuity of the Basis Function at its Ends

Diffuse basis function

Discrete form of electromagnetic integral equations based on boxcar basis functions

Discrete variational methods basis functions

Distributed universal even-tempered basis set of Gaussian functions

Electronic excited states basis functions

Electronic structure calculations with Gaussian basis functions

Elliptical-type basis functions

Expansion in a basis of orbital wave functions

Explicitly correlated Gaussian functions basis sets

Field Potential and Basis Functions

Finite basis functions

Finite basis functions Gaussian-type orbitals

Finite basis functions Slater-type orbitals

Finite basis functions completeness

Finite set of basis functions

Fixed basis functions

Floating-center basis functions

Four-component basis functions

Fourier Series with Complex Exponential Basis Functions

Fourier series with complex basis functions

Functional variation Gaussian basis functions

Functions for Augmenting Basis Sets

G-Spinor Basis Functions

Gases basis” thermodynamic functions

Gaussian Functions Basis Set Preliminaries Direct SCF

Gaussian basis function

Gaussian basis functions Hartree-Fock energies

Gaussian basis functions contracted

Gaussian basis functions electron correlation effects

Gaussian basis functions many-electron molecules

Gaussian basis functions polyatomic molecules

Gaussian basis functions three-dimensional

Gaussian basis sets polarization functions

Gaussian functions, in basis set

Gaussian primitive functions, even-tempered basis sets

Gaussian-type basis functions

Generating Representations from Basis Functions

Geometric Basis and Functionality

Group Product Functions and the Basis Orbitals

Group from basis functions

Hermite basis functions

Hylleraas-type basis functions

INDEX Diffuse basis function

If the Basis Functions are Somewhere Negative

Integrals over basis functions

Introduction to Radial Basis Functions

Inverse problems basis functions, choice

Kernels Radial basis function

Large component Gaussian 2-spinor basis functions

Linear dependence of basis functions

Liquids basis” thermodynamic functions

Many-electron basis functions

Minimal-basis-set wave function

Molecular Basis Sets of Gaussian-type Functions

Molecular function basis expansion

Molecular orbitals basis function expansion

Molecular quantum similarity basis functions

Neglecting a Basis Function

Neural network with radial basis functions

Neural radial basis function

Normalization basis functions

Notation for Gaussian-type basis functions

Nuclear Spin Basis Functions

Numerical basis function

Numerical basis functions anion

Numerical methods basis function

Off-Diagonal Matrix Elements of Total Hamiltonian between Unsymmetrized Basis Functions

One-particle basis functions

Orthogonalization basis functions

Orthonormality basis functions

Other Functional Series with Orthogonal Basis Sets

Parameter-dependent basis functions

Particle basis functions

Phase factor from basis function symmetry

Plane Waves and Atomic-like Basis Sets. Slater-type Functions

Plane wave basis functions

Polarisation/polarisable basis functions

Polynomial basis functions

Population Analysis Based on Basis Functions

Population analysis basis functions

Potential functions theoretical basis

Primitive Cartesian Gaussian basis functions

Primitive basis function Gaussian

Primitive basis function matrix elements

Primitive basis functions

Product basis functions

Quantum chemical equations particle basis functions

Radial Basis Function (RBF)

Radial Basis Function Interpolator

Radial basis function , artificial

Radial basis function collocation

Radial basis function nets

Radial basis function network training

Radial basis function networks

Radial basis function networks (RBF

Radial basis function neural network RBFNN)

Radial basis function neural networks

Radial basis function technique

Representations and Basis Functions

Representations from basis functions

Requirements on one-electron basis functions

Rotational basis functions

Rumer basis, spin functions

Shells of basis functions

Single coordinate basis functions

Space of basis functions

Special Choices of Basis Functions

Special basis functions

Spherical Gaussian basis functions

Spinor Basis Function

Structural Basis of Ion Channel Function

Sturmian basis functions

Symmetry Properties of Hunds Case (a) Basis Functions

Symmetry Properties of non-Hunds Case (a) Basis Functions

Symmetry-adapted basis functions

Symmetry-adapted linear combinations basis functions

System identification basis function

The Basis Function

The Basis State Functions

Thermodynamic functions basis

Tight-binding method basis functions

Time-dependent basis functions

Total angular momentum basis functions

Transfer state function, constructing with basis functions

Two-electron integrals over basis functions

Types of Basis Functions

Wave function analysis basis functions

Wave function basis set

Wavelet basis functions

Weight basis probability density function

What Basis Functions Do

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