Molecular orbitals in momentum space

1965 ◽  
Vol 286 (1406) ◽  
pp. 376-389 ◽  
Keyword(s):  
Momentum Space ◽  
Wave Functions ◽  
Basis Set ◽  
Equilibrium Distance ◽  
Numerical Work ◽  
Algebraic Expressions ◽  
Unitary Transformations ◽  
The Matrix ◽  
Internuclear Distances ◽  
Ground State Energies

The troublesome problem of developing cusps in ordinary molecular wave functions can be avoided by working with momentum-space wavefunctions for these have no cusps. The need for continuum wavefunctions can be eliminated if one works with a hydrogenic basis set in Fock’s projective momentmn space. This basis set is the set of R 4 spherical harmonics and as a consequence one may obtain, solely by the ordinary angular momentum calculus, algebraic expressions for all the integrals required in the solution of the momentum space Schrödinger equation. A number of these integrals and a number of R 4 transformation coefficients are tabulated. The method is then applied to several simple united-atom and l.c.a.o. wavefunctions for H + 2 and ground state energies and corrected wavefunctions are obtained. It is found in this numerical work that the method is most appropriate at internuclear distances somewhat less than the equilibrium distance. In Fock’s representation both l.c.a.o. and unitedatom approximations become exact as the internuclear distance approaches zero. The united-atom expansion can be viewed as an eigenvalue equation for the root-mean-square momentum, p 0 = √( — 2 E ). In the molecule, the matrix operator corresponding to p 0 is related to the operator for the united-atom by a sum of unitary transformations, one for each nucleus in the molecule.

Bloch and Wannier functions in momentum space

1988 ◽  
Vol 53 (9) ◽  
pp. 1890-1901 ◽  
Author(s):  
Jean-Louis Calais
Keyword(s):  
Momentum Space ◽  
Plane Waves ◽  
Wave Functions ◽  
Basis Set ◽  
Wannier Functions ◽  
Atomic Orbitals ◽  
Formal Properties ◽  
Direct Calculations

Direct calculations of wave functions in momentum space require a study of the formal properties of such functions. Here we discuss the momentum space counterparts of Bloch and Wannier functions, both in general and for two extreme kinds of basis set expansions: plane waves and atomic orbitals.

Note on the wave functions used for point-ion-lattice calculations of F-centres

1966 ◽  
Vol 19 (12) ◽  
pp. 2193
Author(s):  
CK Coogan ◽  
HG Hecht
Keyword(s):  
Ground State ◽  
Hyperfine Coupling ◽  
Spherical Wave ◽  
Alkali Halides ◽  
Higher Order ◽  
Wave Functions ◽  
Crystal Symmetry ◽  
Basis Set ◽  
Slater Type ◽  
Ground State Energies

Slater type ns wave functions, differing from the spherically symmetrical wave functions used by Gourary and Adrian, have been tried as a basis set for calculating wave functions of electrons in F-centres in alkali halides. Combinations of 1s, 2s, and 3s Slater functions still yielded ground state energies slightly higher than that calculated by Gourary and Adrian. It is concluded both that the GA wave functions were very well chosen and that more significant changes, particularly in the calculated hyperfine coupling, would come from adding terms of higher-order harmonics, compatible with the crystal symmetry, to the spherical wave functions.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Multiply Separated Synthesis of Six-Link Mechanisms Using Parallel Homotopy With M-Homogenization

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
M. Zhang
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Parallel Implementation ◽  
Homotopy Methods ◽  
Numerical Work ◽  
Function Generation ◽  
Connection Machine ◽  
The Matrix ◽  
Generation Problem ◽  
Complete Solutions

Abstract This paper presents complete solutions to the function generation problem of six-link Watt and Stephenson mechanisms, with multiply separated precision positions (PP), using homotopy methods with m-homogenization. It is seen that using the matrix method for synthesis, applying m-homogeneous group theory and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked in obtaining all possible solutions to the synthesis problem can be drastically reduced. Numerical work dealing with the synthesis of Watt and Stephenson mechanisms for 6 and 9 multiply separated precision points is presented. For both mechanisms, it is seen that complete solutions for 6 and 9 precision points can be obtained by tracking 640 and 286,720 paths, respectively. A parallel implementation of homotopy methods on the Connection Machine on which several thousand homotopy paths can be tracked concurrently is also discussed.

Basis-Set Expansions of Relativistic Electronic Wave Functions

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Relativistic Quantum ◽  
Finite Basis ◽  
Wave Functions ◽  
Basis Set ◽  
Electronic Wave ◽  
Nonrelativistic Case ◽  
Hartree Fock ◽  
Electronic Wave Functions ◽  
Basis Set Expansion ◽  
Molecular Calculations

There have been several successful applications of the Dirac–Hartree–Fock (DHF) equations to the calculation of numerical electronic wave functions for diatomic molecules (Laaksonen and Grant 1984a, 1984b, Sundholm 1988, 1994, Kullie et al. 1999). However, the use of numerical techniques in relativistic molecular calculations encounters the same difficulties as in the nonrelativistic case, and to proceed to general applications beyond simple diatomic and linear molecules it is necessary to resort to an analytic approximation using a basis set expansion of the wave function. The techniques for such calculations may to a large extent be based on the methods developed for nonrelativistic calculations, but it turns out that the transfer of these methods to the relativistic case requires special considerations. These considerations, as well as the development of the finite basis versions of both the Dirac and DHF equations, form the subject of the present chapter. In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basisset selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. There are two basically different strategies that may be followed when developing a finite basis formulation for relativistic molecular calculations. One possibility is to expand the large and small components of the 4-spinor in a basis of 2-spinors. The alternative is to expand each of the scalar components of the 4-spinor in a scalar basis. Both approaches have their advantages and disadvantages, though the latter approach is obviously the easier one for adapting nonrelativistic methods, which work in real scalar arithmetic.

scholarly journals DEVELOPMENT OF MOLECULAR IMPRINTED POLYMER SOLID PHASE EXTRACTION (MISPE) FOR SEPARATION NITROFURANTOIN RESIDUE IN CHICKEN EGGS

2017 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Sophi Damayanti ◽  
Untung Gunawan ◽  
Slamet Ibrahim
Keyword(s):  
Solid Phase Extraction ◽  
Analytical Methods ◽  
Density Functional ◽  
Solid Phase ◽  
Basis Set ◽  
Phase Extraction ◽  
Molecular Imprinted Polymer ◽  
Imprinted Polymer ◽  
Dft Methods ◽  
The Matrix

Background: The use of nitrofurantoin and other nitrofuran antibiotics in food which produced from animals is prohibited by European Union because of potentially carcinogenic and mutagenic. Various methods for analysis of residues of nitrofurantoin has been developed, but because of the interference of the matrix, it is necessary to separate the matrix therefore, the matrix effect will not interfere the analysis. Nowadays, molecular imprinted polymer (MIP) is a well-developed tool in the analytical field, mainly for separating substances in relatively complex matrices.Objective: The purpose of this study is to obtain MISPE that is selective for the separation of nitrofurantoin residues in chicken eggs.Methods: Analytical methods development of nitrofurantoin were optimization of HPLC system and validation of analytical methods performed to obtain the suitable system for nitrofurantoin detection. In silico study used for MIP design by observing the difference Gibbs free energy using Gaussview 5.08 software with Density Functional Theory (DFT) methods using 6-311G as basis set. MIP synthesis was done using bulk method use nitrofurantoin as template, acrylamide as functional monomer, ethyleneglycoldimethacrylate (EGDMA) as crosslinker, and azobisisobutyronitrile (AIBN) as an initiator reaction inside dimethylformamide (DMF) as solvent. Non imprinted polymer (NIP) was synthesized as comparison. MIP and NIP which has been synthesized was inserted into SPE cartridge and characterized using Infrared spectroscopy and HPLC.Result: MISPE that has been synthesized was characterized and compared to non-imprinted polymer solid phase extraction (NISPE) and marketed Solid Phase Extraction (SPE) C18. Sensitivity of MIP, NIP, and SPE C-18 to nitrofurantoin was 84.54 %, 37.73 %, and 33.95 % respectively, based on recovery of nitrofurantoin.Conclusion: Based on the result it was obtained MISPE has high selectivity toward nitrofurantoin compared to NISPE and either marketed SPE.  

TWO-ELECTRON SYSTEM IN THE FIELD OF GENERALIZED SCREENED POTENTIAL

2011 ◽  
Vol 25 (19) ◽  
pp. 1619-1629 ◽  
Author(s):  
ARIJIT GHOSHAL ◽  
Y. K. HO
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Electron System ◽  
Wave Functions ◽  
Expectation Values ◽  
Screening Parameter ◽  
System Ground State ◽  
Highly Correlated ◽  
First Time ◽  
Ground State Energies

Ground states of a two-electron system in generalized screened potential (GSP) with screening parameter λ: [Formula: see text] where ∊ is a constant, have been investigated. Employing highly correlated and extensive wave functions in Ritz's variational principle, we have been able to determine accurate ground state energies and wave functions of a two-electron system for different values of the screening parameter λ and the constant ∊. Convergence of the ground state energies with the increase of the number of terms in the wave function are shown. We also report various geometrical expectation values associated with the system, ground state energies of the corresponding one-electron system and the ionization potentials of the system. Such a calculation for the ground state of a two-electron system in GSP is carried out for first time in the literature.

scholarly journals Chaos from massive deformations of Yang-Mills matrix models

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
K. Başkan ◽  
S. Kürkçüoğlu ◽  
O. Oktay ◽  
C. Taşcı
Keyword(s):  
Gauge Theory ◽  
Time Dependence ◽  
Chaotic Dynamics ◽  
Upper Bounds ◽  
Numerical Work ◽  
Yang Mills ◽  
The Matrix ◽  
Largest Lyapunov Exponents ◽  
Bosonic Part ◽  
Mass Deformation

Abstract We focus on an SU(N ) Yang-Mills gauge theory in 0 + 1-dimensions with the same matrix content as the bosonic part of the BFSS matrix model, but with mass deformation terms breaking the global SO(9) symmetry of the latter to SO(5) × SO(3) × ℤ2. Introducing an ansatz configuration involving fuzzy four and two spheres with collective time dependence, we examine the chaotic dynamics in a family of effective Lagrangians obtained by tracing over the aforementioned ansatz configurations at the matrix levels $$ N=\frac{1}{6} $$ N = 1 6 (n + 1)(n + 2)(n + 3), for n = 1, 2, · · · , 7. Through numerical work, we determine the Lyapunov spectrum and analyze how the largest Lyapunov exponents(LLE) change as a function of the energy, and discuss how our results can be used to model the temperature dependence of the LLEs and put upper bounds on the temperature above which LLE values comply with the Maldacena-Shenker-Stanford (MSS) bound 2πT , and below which it will eventually be violated.

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