Atomic basis functions

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

What guidance for improving the scattering formalism can be obtained from theory In the linear combination of atomic orbitals (LCAO) formalism, a molecular orbital (MO) is described as a combination of atomic basis function ... 

Figure 2 The one-electron energies = -1/(26 ) for an electron moving in the field of two protons are shown here as functions of the internuclear distance R. The energies represented by the smooth curves were calculated using 10 atomic basis functions on each center. For the lowest curve, the best available calculated values [22] are indicated by dots. All energies are in Hartrees. 

The molecular four-component spinors constructed this way are orthogonal to the inner core spinors of the atom, because the atomic basis functions used in Eq. (6.4) are generated with the inner core shells treated as frozen. 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

Atomic basis functions on B are straightforward to classify. Evidently, an s type function on B will be totally symmetric — an a orbital. A quick inspection of the D3h character table will show that a p set on B, which transforms like the three Cartesian directions, spans the reducible representation a 2 e. Functions centred on the F atoms require more effort. Since the operations in the classes containing C3 and S3 move all three F atoms, their character is necessarily zero for any functions centred on the F atoms. Consider first a set of s functions on each F. These span a reducible representation with character... 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

We can go beyond the classification of atomic basis functions of the last subsection to obtain explicit symmetry-adapted basis functions by projection. We can use either the full matrix projection and shift operators (Eqs. 1.28 and 1.31, which sure repeated here for convenience) ... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

Day21 has given a careful account of the relationship between the Woodward-Hoffmann rules and Mobius/Hiickel aromaticity, and has defined the terms supra-facial and antarafacial in terms of the nodal structure of the atomic basis functions. His approach makes quite explicit the assumption that the transition state involves a cyclic array of basis functions. Thus the interconversion of prismane (10) and benzene, apparently an allowed (n2s+ 2S+ 2S) process, is in fact forbidden because there are additional unfavourable overlaps across the ring.2... 

An ab initio calculation is defined by two parameters the (atomic) basis functions (or basis sets) employed and the level of electron correlation adopted. These two topics will be described in some detail (and qualitatively) in the next two sections. 

Indicating with % and % the atomic basis functions centred on the moieties A (with Na electrons) and B (with Nt electrons), respectively, the SC orbitals are expanded in the form ... 

The approximations used in EH theory have included (1) neglect of core electrons, (2) use of atomic basis functions, (3) use of effective Hamiltonian resulting in arbitrariness in choice of matrix elements, (4) lack of explicit accounting for electron-electron and nuclear-nuclear repulsion, and (5) approximate energy calculation procedure. Although these approximations seems severe in light of Hartree-Fock treatments of MO theory, the important feature to remember is that EH theory has proved useful in several systems. 

The computer time required for ab initio calculations is roughly proportional to the fourth power of the number of atomic basis functions used for the description of the molecular system. Ab initio calculations are thus not feasible today for host-guest systems with more than about 150—200 electrons. Supercomputers and vector processors will significantly lower the necessary CPU times150) but they alone probably cannot bring a breakthrough for systems larger than two or three times the ones which can be treated today. 

In other words, P(A,B z) measures the probability that the electron occupying Xi will be detected in the diatomic fragment AB of the molecule. The inequality in the preceding equation reflects the fact that the atomic basis functions participate in chemical bonds with all constituent atoms, with the equality sign corresponding only to a diatomic molecule, when /AB = /. ... 

In the early 1950s, Roothaan43 [15] and Hall44 [16] independently suggested that one expand the N molecular orbitals (MO) wavefimctions cj), in the chosen basis set as linear combinations of a complete set of B atomic basis functions... 

Equation (29) is an analog of Eq. (5) for the case of atomic basis functions represented as two-component spinors. 

Let s turn our attention to the computational needs of the structure problem (See Fig. 2). Defining n as the number of atomic basis functions employed, there are several characteristic matrices to consider. The Fock matrix, which we repeatedly construct and diagonalize until convergence, is only an n x n matrix unfortunately, to construct this matri at each iteration of the SCF procedure, we have to process the n /8 two-electron integrals. At the current state of the art, the number of atomic basis functions n tends to be about 100. This limitation is not so much because of the difficulty of diagonalizing 100 x 100 matrices, but because of the 10 limitations inherent with processing the tens of millions of integrals at each iteration. 

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