Schrodinger equation mathematical methods

We have seen three broad techniques for calculating the geometries and energies of molecules molecular mechanics (Chapter 3), ab initio methods (Chapter 5), and semiempirical methods (Chapters 4 and 6). Molecular mechanics is based on a balls-and-springs model of molecules. Ab initio methods are based on the subtler model of the quantum mechanical molecule, which we treat mathematically starting with the Schrodinger equation. Semiempirical methods, from simpler ones like the Hiickel and extended Hiickel theories (Chapter 4) to the more complex SCF semiempirical theories (Chapter 6), are also based on the Schrodinger equation, and in fact their empirical aspect comes from the desire to avoid the mathematical... 

We need mathematical methods which will allow us to obtain approximate solutions of the Schrodinger equation. These methods are the variational method and the perturbational approach. 

Z. A. Anastassi and T. E. Simos, Trigonometrically Fitted Fifth Order Runge-Kutta Methods for the Numerical Solution of the Schrodinger Equation, Mathematical and Computer Modelling, 2005, 42(7-8), 877-886. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

For any but the smallest systems, however, exact solutions to the Schrodinger equation are not computationally practical. Electronic structure methods are characterized by their various mathematical approximations to its solution. There are two major classes of electronic structure methods ... 

Ab initio methods compute solutions to the Schrodinger equation using a series of rigorous mathematical approximations. These procedures are discussed in detail in Appendix A, The Theoretical Background. 

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

The term "semi-empirical" has been reserved commonly for electronic-based calculations which also starts with the Schrodinger equation.9-31 Due to the mathematical complexity, which involve the calculation of many integrals, certain families of integrals have been eliminated or approximated. Unlike ab initio methods, the semi-empirical approach adds terms and parameters to fit experimental data (e.g., heats of formation). The level of approximations define the different semi-empirical methods. The original semi-empirical methods can be traced back to the CNDO,12 13 NDDO, and INDO.15 The success of the MINDO,16 MINDO/3,17-21 and MNDO22-27 level of theory ultimately led to the development of AMI28 and a reparameterized variant known as PM3.29 30 In 1993, Dewar et al. introduced SAMI.31 Semi-empirical calculations have provided a wealth of information for practical applications. 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

In the ab initio approach the desired answers are the experimental observables - spectral line positions, shapes, intensities scattering and reaction rates polarizabilities and optical rotary power etc. These are to be obtained from the Schrodinger equation by numerical methods which are mathematically well-defined and involve no intermediate parameters not appearing in the Schrodinger equation itself. 

Density functional calculations (DFT calculations, density functional theory) are, like ab initio and semiempirical calculations, based on the Schrodinger equation However, unlike the other two methods, DFT does not calculate a conventional wavefunction, but rather derives the electron distribution (electron density function) directly. Afunctional is a mathematical entity related to a function. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

The electron distribution around an atom can be represented in several ways. Hydrogenlike functions based on solutions of the Schrodinger equation for the hydrogen atom, polynomial functions with adjustable parameters, Slater functions (Eq. 5.95), and Gaussian functions (Eq. 5.96) have all been used [34]. Of these, Slater and Gaussian functions are mathematically the simplest, and it is these that are currently used as the basis functions in molecular calculations. Slater functions are used in semiempirical calculations, like the extended Hiickel method (Section 4.4) and other semiempirical methods (Chapter 6). Modem molecular ab initio programs employ Gaussian functions. 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an atom in molecule as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus ... 

T. E. Simos, A Family of Four-Step Trigonometrically-Fitted Methods and its Application to the Schrodinger Equation, Journal of Mathematical Chemistry, in press. 

The particle in a box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the three-dimensional Schrodinger equations. The same methods used for the one-dimensional box can be expanded to three dimensions for atoms. These orbital equations include three... 

Quantum effects can be recovered by quantum simulations. Currently there are two main types of quantum simulation methods used. One is based on the time-dependent Schrodinger equation. The other is based on Feynman s path integral (PI) quantum statistical mechanics. [7,8] The former is usually complicated in mathematical treatment and needs also large computational resources. Currently, it can only be used to simulate some very limited systems. [77] MD simulations based on the latter have been used more than a decade and are gaining more and more popularity. The main reason is that in PIMD simulations, the quantum systems are mapped onto corresponding classical systems. In other words the quantum effects can be recovered by making a series of classical simulations with different effective potentials. 

Methodology. Unquestionably, the application of quantum mechanics to chemical bonding has revolutionized scientific thinking. In fact, the modern theoretical framework of chemistry rests on quantum physics. In principle, the Schrodinger equation may be solved for any chemical system. No prior knowledge of any analogous or related system is necessary. Exactly solvable problems are rare, due to the mathematical complexities recourse must then be made to approximate methods, and many powerful approaches have been devised. Generally, approximate solutions must suffice for the size of molecules of pharmaceutical interest. 

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