Gaussian basis

Jordan oompared the use of plane wave and oonventional Gaussian basis orbitals within density funotional oaloulations in ... 

Dunning T FI Jr 1970 Gaussian basis funotions for use in moleoular oaloulations I. Contraotion of (9s 5p) atomio basis sets for the first-row atoms J. Chem. Phys. 53 2823-33... 

Dunning T FI Jr and Flay P J 1977 Gaussian basis sets for moleoular oaloulations Methods of Electronic Structure Theory vo 3, ed FI F III Sohaefer (New York Plenum) pp 1-27... 

Flelgaker T and Taylor P R 1995 Gaussian basis sets and molecular integrals Modem Electronic Structure Theory yo 2, ed D R Yarkony (Singapore World Scientific) section 5.4, pp 725-856... 

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

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... 

Bade Z and Light J C 1986 Highly exdted vibrational levels of floppy triatomic molecules—a discrete variable representation—distributed Gaussian-basis approach J. Chem. Phys. 85 4594... 

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... 

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

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... 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In addition to the mixed results in Table 10-1, the G2 calculation for H2 produces an energy that is lower than the experimental value, in contradiction to the rule that variational procedures reach a least upper bound on the energy. Some new factors are at work, and we must look into the stmcture of the G2 procedure in temis of high-level Gaussian basis sets and electron correlation. 

COMPUTER PROJECT 10-1 Gaussian Basis Sets The HF Limit... 

In hybrid DET-Gaussian methods, a Gaussian basis set is used to obtain the best approximation to the three classical or one-election parts of the Schroedinger equation for molecules and DET is used to calculate the election correlation. The Gaussian parts of the calculation are carried out at the restiicted Hartiee-Fock level, for example 6-31G or 6-31 lG(3d,2p), and the DFT part of the calculation is by the B3LYP approximation. Numerous other hybrid methods are currently in use. 

The values of the orbital exponents ( s or as) and the GTO-to-CGTO eontraetion eoeffieients needed to implement a partieular basis of the kind deseribed above have been tabulated in several journal artieles and in eomputer data bases (in partieular, in the data base eontained in the book Handbook of Gaussian Basis Sets A. Compendium for Ab initio Moleeular Orbital Caleulations, R. Poirer, R. Kari, and I. G. Csizmadia, Elsevier Seienee Publishing Co., Ine., New York, New York (1985)). 

Each of these factors can be viewed as combinations of CSFs with the same Cj and Cyj coefficients as in F but with the spin-orbital involving basis functions that have been differentiated with respect to displacement of center-a. It turns out that such derivatives of Gaussian basis orbitals can be carried out analytically (giving rise to new Gaussians with one higher and one lower 1-quantum number). 

R. Poirer, R. Kari, I. G. Csizmadia, Handbook of Gaussian Basis Sets A Compendium for Ah Initio Molecular Orbital Calculations Elsevier Science Publishing, New York (1985). 

Gaussian basis sets for molecular calculations S. Huzinaga, Ed., Elsevier, Amsterdam (1984). 

Chapter 10 represented a wave function as a linear combination of Gaussian basis functions. Today, there are so many basis sets available that many researchers will never need to modify a basis set. However, there are occasionally times when it is desirable to extend an existing basis set in order to obtain more accurate results. The savvy researcher also needs to be able to understand the older literature, in which basis sets were customized routinely. 

McLean, A.D. Chandler, G.S. Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z=ll-18 J. Chem. Phys. 72 5639-5648, 1980. 

Gaussian Basis Functions for use in Molecular Calculations III Contraction of (10s, 6p) Atomic Basis Sets for the First-Row Atoms T. FI. Dunning, Jr... 

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