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142 Peter M W Gill,Table of Contents,1 Quantum Chemical Procedures. 2 Basis Functions,2 1 Slater Functions,2 2 Gaussian Functions. 2 3 Contracted Gaussian Functions,2 4 Gaussian Lobe Functions. 2 5 Delta Functions,3 A Survey of Gaussian Integral Algorithms. 3 1 Performance Measures,3 1 1 mop Cost,3 1 2 Mop Cost.

3 1 3 CPU Time,3 2 Fundamental Integrals,3 2 1 The Overlap Integral. 3 2 2 The Kinetic Energy Integral,3 2 3 The Electron Repulsion Integral. 3 2 4 The Nuclear Attraction Integral,3 2 5 The Anti Coulomb Integral. 3 3 The Boys Algorithm,3 4 The Contraction Problem. 3 5 The Pople Hehre Algorithm,3 6 Bras Kets and Brakets.

3 7 The McMurchie DavidsonAlgorithm,3 8 The Obara Saika SchlegelAlgorithm. 3 9 The Head Gordon Pople Algorithm,3 10 Variations on the HGP Theme. 4 The PRISM Algorithm,4 1 Shell Pair Data,4 2 Selection of Shell Quartets. 4 3 Generation of the O m Integrals,4 4 Contraction Steps. 4 5 Transformation Steps, 4 5 1 Two Electron Transformations on the MD PRISM.

4 5 2 One Electron Transformations on the MD PRISM. 4 5 3 Two Electron Transformations on the HGP PRISM. 4 5 4 One Electron Transformations on the HGP PRISM. 4 6 Loop Structure of PRISM in Gaussian 92,4 7 Performance of PRISM in Gaussian 92. 5 Prospects for the Future,6 Acknowledgments,7 References. Molecular Integrals over Gaussian Basis Functions 143. 1 QUANTUM CHEMICAL PROCEDURES, The major goal of Quantum Chemistry is to obtain solutions to atomic and. molecular Schrodinger equations l To be useful to chemists such solutions. must be obtainable at a tolerable computational cost and must be reasonably. accurate yet devising solution methods which meet both of these requirements. has proven remarkably difficult Indeed although very many variations have. been developed over the years almost every currently existing method can be. traced to a prototype introduced within ten years of Schrodinger s seminal paper. The most uniformly successful family of methods begins with the simplest. possible n electron wavefunction satisfying the Pauli antisymmetry principle a. Slater determinant 2 of one electron functions x i r w called spinorbitals Each. spinorbital is a product of a molecular orbital yi r and a spinfunction a o. or p o The y i r are found by the self consistent field SCF procedure. introduced 3 into quantum chemistry by Hartree The Hartree Fock HF 4. and Kohn Sham density functional KS 5 6 theories are both of this type as are. their many simplified variants 7 161, In SCF methods the n electron Schrodinger equation is replaced by a set. of n coupled integro differential one electron equations The HF equations are a. well defined approximation to the Schrodinger equation and constitute the starting. point for a variety of subsequent treatments 171 the equations of KS theory are. formally equivalent 181 to the original Schrodinger equation for the ground state. In both cases the equations are highly non linear and require iterative techniques. for their solution, Commonly initial guesses for the molecular orbitals are obtained and.

these are then used to compute the potential felt by an electron in the field of the. nuclei and the other electrons The corresponding one electron Schrodinger. equation is then solved to determine another set of orbitals and the process is. continued until successive sets of orbitals differ negligibly at which point self. consistency is said to have been achieved The most time consuming part of this. procedure is the evaluation of the potential which within a basis set Section 2. is represented by various types of integrals Section 3 Moreover even if we. proceed beyond the HF SCF level to correlated levels of theory these integrals. remain central to the problem of determining the energy and wavefunction 171. 144 Peter M W Gill, The result of any quantum chemical procedure is the molecular energy. parametrically determined by the nuclear geometry To locate equilibrium and. transition structures we usually compute the first derivatives of the energy with. respect to nuclear motion 191 harmonic vibrational frequencies can be obtained. if second derivatives are available 20 third and higher derivatives are. needed 21 for higher level studies of potential surfaces Not surprisingly nth. derivatives of the integrals are required to compute nth derivatives of the energy. and the efficient generation of integrals and their nth derivatives is the focus of. this Review,2 BASIS FUNCTIONS, Because computers can represent numbers but not functions the. molecular orbitals at each stage of the SCF procedure have to be represented by. an expansion in a finite set of basis functions qi r i 1 2 N If the set is. mathematically complete the result of the SCF procedure is termed the HF or KS. limit otherwise the result is dependent on the basis set used Many types of basis. funtion have been explored and several are currently used in routine applications. However their interrelationships and relative strengths and weaknesses are not. often clarified and it may be instructive to do so here. 2 1 Slater Functions,Until the 1960 s Slater basis functions 22. later r x A X aAY z A a z e p a l r A l 1, were very popular Like exact wavefunctions they exhibit cusps at the nuclei and. decay exponentially but their use necessitates the evaluation of integrals which are. very time consuming to compute Although several groups have made useful. progress in developing efficient algorithms for the evaluation of such integrals. explicit use of Slater basis functions is presently restricted to rather small. molecules It should be noted however that any Slater function can be. approximated to any desired accuracy by a sum of Gaussian functions and the. difficult Slater integrals then become relatively easy contracted Gaussian integrals. see below This is the philosophy of the STO nG basis sets 23 In a similar. vein the product of a pair of Slater functions can also be approximated to any. accuracy by a sum of Gaussians and this approach has been suggested and. explored by Harris and Monkhorst 24, Molecular Integrals over Gaussian Basis Functions 145.

2 2 Gaussian Functions,A primitive Gaussian function. has center A Ax A A angular momentum a ax a a and exponent a. The suggestion by Boys 25 to use Gaussians as basis functions was a crucial. step in the progression of quantum chemistry from a largely qualitative to a. largely quantitative discipline The use of a Gaussian basis set in a HF or KS. calculation leads to very much simpler integrals see below than those which. arise within a Slater basis and although it is known 26 that more Gaussian than. Slater functions are required to achieve a given basis set quality the simplicity of. Gaussian integrals more than compensates for this, A set of primitive basis functions with the same center and exponent are. known as a primitive shell For example a set of p functions pn pr pz on an. atom is termed a primitive p shell and if an s function with the same exponent. is added the shell becomes a primitive sp shell The most commonly occuring. shells in modern computational chemistry are s p sp d and. 2 3 Contracted Gaussian Functions, It is found that contracted Gaussian functions CGFs 27. where KA is the degree of contraction and the D d are contraction coefiicients are. even more computationally effective 26 than Slater functions It is crucial to. note that although they have different contraction coefficients and exponents all. of the primitive functions in a contracted function share the same center A and. angular momentum a A set of CGFs with the same center and the same set of. exponents is termed a contracted shell by analogy with a primitive shell defined. 146 Peter M W Gill, Over the years many contracted Gaussian basis sets have been constructed. and the interested reader will find the excellent review by Davidson and. Feller 26 very illuminating As a rule one or two CGFs are used to model each. of the core atomic orbitals Is for lithium to neon Is 2s and 2p for sodium to. argon erc and the CGFs are often highly contracted a typical K value is 6. Each valence atomic orbital Is for hydrogen and helium 2s and 2p for lithium to. neon erc is generally more weakly contracted K less than about 4 Finally. high quality basis sets contain functions whose angular momenta are higher than. that of the valence orbitals e g p for hydrogen and helium d for lithium to argon. erc and in most cases these functions are uncontracted K 1. Two distinct classes of contracted Gaussian functions are in common use. In general contraction schemes different contracted functions share the same. primitive exponents with different contraction coefficients while in segmented. schemes different contracted functions are constructed from primitive functions. with different exponents As a rule basis functions of the former type tend to. have higher degrees of contraction but the higher computational cost implied by. this can be partially ameliorated by the use of algorithms which are carefully. constructed to take maximum advantage of the exponent sharing In this Review. we will confine our attention to the efficient treatment of segmented basis sets. we will extend our analysis to the generally contracted case in a future paper. 2 4 Gaussian Lobe Functions, Many of the programming complexities which arise when general.

contracted Gaussian functions are used disappear if all of the functions are. constrained to be s functions i e,r D e p uklr A1 4. Such basis functions were advocated by a number of authors 28 on the basis of. their manifest simplicity and because an array of variously centered s functions. can mimic functions of higher angular momentum d f etc However for. obvious reasons Gaussian Lobe basis sets have to be rather large to yield useful. results and become unwieldy in high angular momentum cases They are rarely. used nowadays because of the availability of highly efficient algorithms and. programs which can handle CGFs of arbitrary angular momentum. Molecular Integrals over Gaussian Basis Functions 147. 2 5 Delta Functions, Still more of the programming complexities vanish if another constraint is. applied to the Gaussian basis set namely that its exponents be infinite which. yields a basis composed entirely of Dirac delta functions. The set of delta functions at all points in space is mathematically complete and. procedures based on these simplest of basis functions have been devised and. implemented first 29 331 for diatomics and more recently 34 361 for arbitrary. polyatomic systems, The manifest simplicity of delta functions is both their strength and. weakness computer programs based on them are refreshingly straightforward. but to yield results of chemical significance delta basis sets must be large. typically thousands of functions per atom The construction of efficient delta. basis sets i e 3 dimensional grids remains an active area of research but most. commonly they consist of points on concentric spheres about each atom Most. workers use the results of Lebedev 37 who has found optimal quadrature. formulae for the surface of a sphere However agreement has not yet been. reached on which spherical radii are best 38 42,I 4a Peter M W Gill. 3 SURVEY OF GAUSSIAN INTEGRAL ALGORITHMS, What are these integrals to which we have referred From the fact that.

the Schrodinger Hamiltonian contains only one and two electron operators it is. straightforward to show 171 that most of the matrix elements 43 which arise in. computing the SCF energy and its derivatives with respect to nuclear motion can. be written in terms of integrals of the general form. and their nth derivatives with respect to displacement of the basis functions Each. of the integrations in 6 is over 3 dimensional space and thus the integral is. 6 dimensional The function f x is normally very simple for example. f x E l x in the familiar case of two electron repulsion integrals but it suffices. for our present purposes to consider a general function. The integral 6 is based on two pairs of basis functions one describing. electron 1 and the other describing electron 2 Since there are N functions in the. basis set there are N N 1 2distinct basis function pairs and similarly there are. 1 N N l N N l 1 81N N 1 N 2 N 2 7, distinct integrals of the form 6 A class of integrals and or their nth derivatives. is defined as the set of all integrals associated with a shell quartet For example a. pplpp class is the set of 8 1 pplpp integrals associated with four p shells each. containing three p functions Because all of the integrals in a class share the same. four centers and sets of exponents their generation involves many common. intermediate quantites For this reason it is always computationally expedient to. compute integrals and their derivatives in classes rather than individually. When large basis sets N 10 are used the generation of the integrals is a. major computational task in fact in the most common quantum chemistry. methods such as direct SCF either in the context of H F 44 or KS 45. calculations it is rate determining Obviously therefore it is of paramount. importance to devise and implement highly efficient generation algorithms This. realization has stimulated the development of a series of integral strategies. Molecular Integrals over Gaussian Basis Functions 143 1 QUANTUM CHEMICAL PROCEDURES The major goal of Quantum Chemistry is to obtain solutions to atomic and molecular Schrodinger equations l To be useful to chemists such solutions must be obtainable at a tolerable computational cost and must be reasonably accurate yet devising solution methods which meet both of these requirements has

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