Method, semi-empirical

Semi-empirical models begin with the HF and LCAO approximations resulting in the Roothaan-HaU equations (Equations 9-36 through 940). A minimal basis set is used of STO s. The Roothaan-Hall equations are solved in a self-consistent field fashion, however not aU of the integrals are actually solved. In the most severe approximation, there is complete neglect of differential overlap (CNDO). 

The values of the parameters used in semi-empirical computations, so that their results agree with experiment and that of HF-SCF-LCAO computations with minimal basis sets, must come from either experimental values or computational values much like in the case of molecular mechanics. As a consequence, care must be taken to use these packages only for the type of molecules for which the packages have been parameterized. The MNDO 

Although density-functional methods as outlined in the preceding sections often are considered powerful and although the available computational power has increased enormously during the last decades, the class of systems that can be treated with such methods is limited. When the number of atoms increases above roughly 50 these methods run into severe problems and if one still attempts to use them, one most often has to make approximations that may or may not influence the results in some uncontrollable and undesirable way. For instance, only certain high-symmetry structures may be considered or the basis set may be limited. 

In order to overcome these problems, approximate methods constructed explicitly for certain classes of systems may be very usefiil. In this section we shall explicitly study two such methods, i.e. a tight-binding density-functional method and the embedded-atom method. 

is related to the exchange-correlation potential Kc(r) through 

By assuming that the Kohn-Sham equations (14) are solved exactly, E may also be written as 

Within a tight-binding approximation it is assumed that E, contains inter- 

Semi-empirical quantum chemistry methods are based on the Hartree-Fock formalism, but make many approximations and obtain some parameters from empirical data. They are very important in computational chemistry for treating large molecules where the full Hartree-Fock method without the approximations is too expensive. The use of empirical parameters appears to allow some inclusion of electron correlation effects into the methods. Within the framework of Hartree-Fock calculations, some pieces of information (such as two-electron integrals) are sometimes approximated or completely omitted. 

In order to correct for this loss, semi-empirical methods are parametrized, that is their results are fitted by a set of parameters, normally in such a way as to produce results that best agree with experimental data, but sometimes to agree with ab initio results. Semi-empirical methods follow what are often called empirical methods where the two-electron part of the Hamiltonian is not explicitly included. 

For TT-electron systems, this was the Hiickel method proposed by Erich Hiickel. For all valence electron systems, the extended Hiickel method was proposed by Roald Hoffmann. Semi-empirical calculations are much faster than their ab initio coxmter-parts. Their results, however, can be very wrong if the molecule being computed is not similar enough to the molecules in the database used to parametrize the method. Semi-empirical calculations have been most successful in the description of organic chemistry, where only a few elements are used extensively and molecules are of moderate size. However, semi-empirical methods were also appHed to solids and nanostructures but with different parameterization. As with empirical methods, we can distinguish if  

Restricted to 7t-electrons. These methods exist for the calculation of electronically excited states of polyenes, both cycHc and linear. These methods, such as the Pariser-Parr-Pople method (PPP), can provide good estimates of the 7t-electronic excited states, when parameterized well. Indeed, for many years, the PPP method outperformed ah initio excited state calculations [6]. 

AMI is currently one of the most commonly used of the Dewar-type methods. It was the next semiempirical method introduced by Dewar and coworkers in 1985 following MNDO. It is simply an extension, a modification to and also a reparameterization of the MNDO method. AMI differs from MNDO by mainly two ways. The first difference is the modification of the core repulsion function. The second one is the parameterization of the overlap terms (3s and (3p, and Slater-type orbital exponents (s and (p on the same atom independently, instead of setting them equal as in MNDO. MNDO had a very strong tendency to overestimate repulsions between atoms when they are at approximately their van der Waals distance apart. To overcome this hydrogen bond problem, the net electrostatic repulsion term of MNDO, J RAH) given by equation (8.2), was modified in MNDO/H to be 

The central assumption of semi-empirical methods is the Zero Differential Overlap (ZDO) approximation, which neglects all products of basis functions depending on the same electron coordinates when located on different atoms. Denoting an atomic orbital on centre. A as /ja (it is customary to denote basis functions with /x, u, A and a in semi-empirical theory, while we are using Xn, X6 for ab initio methods), the ZDO 

To compensate for these approximations, the remaining integrals are made into parameters, and their values are assigned on the basis of calculations or experimental data. Exactly how many integrals are neglected, and how the parametetizalion is tlone, defines the various semi-empirical methods. 

Rewriting eq. (3.51) with semi-empirical labels gives the following expression for a Fock matrix element, where a two-electron integral is abbreviated as fiu Xo), (eq. (3.56)).----------------------------------------------------------------------- 

The Hiickel method is a very primitive example of a semi-empirical method in which various integrals are set equal to either a or p and treated as empirical parameters overlap integrals are ignored. The removal of the restriction of the Hiickel method to planar hydrocarbon systems was achieved with the introduction of the extended Hiickel theory (EHT) in about 1963. In heteroatomic non-planar systems (such as d-metal complexes) the separation of orbitals into k and a is no longer appropriate and each type of atom has a different value of Hu (which in Hiickel theory is set equal to a for all atoms). In this approximation, the overlap integrals are not set equal to zero but ue cdculated expKdtly. Furthermore, the Hjk, which in Hiickel theory are set equal to p, in EHT are made proportional to the overlap integral between the orbitals J and K. 

Further approximations of the Hiickel method were removed with the introduction of the complete neglect of differential overlap (CNDO) method, which is a slightly more sophisticated method for dealing with the terms H,k that appear in the secular equations for the coefficients. The introduction of CNDO opened the door to an avalanche of similar but improved methods and their accompanying acronyms, such as intermediate neglect of differential overlap (INDO), modified neglect of differential overlap (MNDO), and the Austin Model 1 (AMI, version 2 of MINDO). Software for all these procedures is now readily available, and reasonably sophisticated calculations can be run even on handheld computers. 

A semi-empirical technique that has gained considerable ground in recent years to become one of the most widely used techniques for the calculation of molecular structure is density functional theory (DFT). Its advantages include less demanding computational effort, less computer time, and—in some cases (particularly d-metal complexes)—better agreement with experimental values than is obtained from other procedures. 

The central focus of DFT is the electron density, p (rho), rather than the wave-function jf. When the Schrodinger equation is expressed in terms of p, it becomes a set of equations called the Kohn-Sham equations. As for the Schrodinger equation itself, this equation is solved iteratively and self-consistently. First, we guess the electron density. For this step it is common to use a superposition of atomic electron densities. Next, the Kohn-Sham equations are solved to obtain an initial set of orbitals. This set of orbitals is used to obtain a better 

In mathematics, when an entire function/(x) is associated with a single number, F, as when an entire wavefunction is associated with the energy of the state, the number is said to be a functional of the function and written P[f]. Thus, the energy is a functional of the wavefunction and we could denote it E[ if] to denote the functional dependence of the energy on the entire wavefunction. In DFT, the energy is regarded as a functional of the electron density, and written E[p]. 

In order to make calculations on large systems possible, e.g. optimization of inter-atomic distances in the types of system that are of interest for metal-polymer interfaces, it is often necessary to work with more approximate theoretical methods. This does not automatically leads to poorer results than from, for instance, a minimal basis set ab initio Hartree-Fock calculation. However, since the method relies on a kind of fitting procedure it is particularly important to make comparison with experimental data since this is the only test of whether this procedure is valid or not. 

The approximation schemes that are discussed in this paragraph are all based on the Hartree-Fock theory, i.e., the many-electron wavefunction is described in terms of a single Slater determinant. 

The semi-empirical approach was first formulated by Pople10. In the first version, the approximation made on the two-electron integrals (see eq. (2. 12))was of the form  

Following the initial work of Pople, there appeared a number of integral approximation schemes, all combined with different sets of empirical parameters. One such approximation scheme that has been 

The solution of eq. (3.79) can be done by repeated diagonalization of the F matrix, analogously to the situation for non-periodic systems. A plane wave basis, however, often involves several thousand functions, which means that alternative methods are used for solving the equation. 

The first step in reducing the computational problem is to consider only the valence electrons explicitly the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third 

ELECTRONIC STRUCTURE METHODS INDEPENDENT-PARTICLE MODELS 

In the decade between 1985 and 1995, molecular orbital calculations using semi-empirical methods for 1,2,3-triazoles and benzotriazoles have received increasing interest. In particular, the semi-empirical methods AMI, PM3, and MNDO, have been widely used in theoretical calculations for 

The reliability of semi-empirical methods (AM 1, PM3, and MNDO) for the treatment of tautomeric equilibria has been tested in a series of five-membered nitrogen heterocycles, including 

3- triazole and benzotriazole. The known tendency of MNDO to overestimate the stability of heterocycles with two or more adjacent pyridine-like lone pairs is also present in AMI and to a somewhat lesser extent in PM3. Tautomeric isomers with a different number of adjacent pyridinelike nitrogens cannot be adequately treated by these semi-empirical methods. Both AMI and PM3 represent major improvements over MNDO in the case of lactam lactim tautomerism. The stability of A-oxides as compared to A-hydroxy tautomers is overestimated by the PM3 method. All three methods give reliable ionization potentials and dipole moments 90ZN(A)1328 . 

Bond lengths (A) Calc. Exp. Bond angles (°) Calc. Exp. 

ASED (Atom Superposition and Electron Delocalization) molecular orbital calculations on the formation of monomeric 1-triazolylborane by the process BH3-I-triazole H2-f H2B(Tz) indicate 

Electronic strucmre methods are characterized by their various mathematical approximations to its solution, since exact solutions to the Schrddinger equation are not computationally practical. There are three classes of electronic structure methods semi-empirical methods, density functional theory (DFT) methods, and 

Many approximate molecular orbital theories have been devised. Most of these methods are not in widespread use today in their original form. Nevertheless, the more widely used methods of today are derived from earlier formalisms, which we will therefore consider where appropriate. We will concentrate on the semi-empirical methods developed in the research groups of Pople and Dewar. The former pioneered the CNDO, INDO and NDDO methods, which are now relatively little used in their original form but provided the basis for subsequent work by the Dewar group, whose research resulted in the popular MINDO/3, MNDO and AMI methods. Our aim will be to show how the theory can be applied in a practical way, not only to highlight their successes but also to show where problems were encountered and how these problems were overcome. We will also consider the Hiickel molecular orbital approach and the extended Hiickel method Our discussion of the underlying theoretical background of the approximate molecular orbital methods will be based on the Roothaan-Hall framework we have already developed. This will help us to establish the similarities and the differences with the ab initio approach. 

A discussion of semi-empirical methods starts most appropriately with the key components 

In ab initio calculations aU elements of the Fock matrix are calculated using Equation (2J226), irrespective of whether the basis fimctions (j , (j x and 4 are on the same atom, on atoms that are bonded or on atoms that are not formally bonded. To discuss the semi-empirical methods it is useful to consider the Fock matrix elements in three groups (the diagonal 

A feature common to the semi-empirical methods is that the overlap matrix, S (in Equation (2.225)), is set equal to the identity matrix I. Thus all diagonal elements of the overlap matrix are equal to 1 and all off-diagonal elements are zero. Some of the off-diagonal elements would naturally be zero due to the use of orthogonal basis sets on each atom, but in addition the elements that correspond to the overlap between two atomic orbitals on different atoms are also set to zero. The main implication of this is that the Roothaan-Hall equations are simplified FC = SCE becomes FC = CE and so is immediately in standard matrix form. It is important to note that setting S equal to the identity matrix does not mean that aU overlap integrals are set to zero in the calculation of Fock matrix elements. Indeed, it is important specifically to include some of the overlaps in even the simplest of the semi-empirical models. 

Many semi-empirical theories are based upon the zero-differential overlap approximation (ZDO). In this approximation, the overlap between pairs of different orbitals is set to zero for all volume elements dv  

Rewriting eq. (3.51) with semi-empirical labels gives the following expres.sion for a Fock matrix element, where a two-electron integral is abbreviated as (LLv a), (eq. (3.56)). 

1 Neglect of Diatomic Differentiai Overiap Approximation (NDDO) 

We first recall that the value pertinent in the electron transfer problem is that evaluated for the nuclear configuration Q Q, where the energy of the interseetion surface of (Q) and Hbb (Q) is a minimum. In some systems, it may happen that vl/ and 1]/ are closely related to stationary states of the Hamiltonian H, so that spectroscopic experiments performed on these states may provide useful information about the value of [47, 48]. To clarify this point, we expand the stationary states )/i (i= 1,2,. . . ) of H(r, Q) in the form  

we assume that this expansion can be safely limited to the first two terms for the ground state vl/j and the first excited state 2- For the nuclear configuration where the energy difference  

This simple model allows the evaluation of Tba(Qb) from the measurement of the exchange parameter J if the energy U is known. 

Returning to the general case, we find that for the nuclear configuration Q Q where and H b are equal, the mixing coeflScients and the energies satisfy the following relationships  

relation (18) gives directly the value of lTab(Q ) as half the difference between the energies of the two stationary states /i and xJ/j, caleulated at the nuclear configuration Q = Q. We shall see in Sect. 2.2.3 some examples of theoretical calculations of the electronic factor which are based on this property. 

While orbitals may be useful for qualitative understanding of some molecules, it is important to remember that they are merely mathematical functions that represent solutions to the Hartree-Fock equations for a given molecule. Other orbitals exist which will produce the same energy and properties and which may look quite different. There is ultimately no physical reality which can be associated with these images. In short, individual orbitals are mathematical not physical constructs. 

Semi-empirical methods may only be used for systems where parameters have been developed for all of their component atoms. In addition to this, semi-empirical models have a number of well-known limitations. Types of problems on which they do not perform well include hydrogen bonding, transition structures, molecules containing atoms for which they are poorly parametrized, and so on. We consider one such case in the following example, and the exercises will discuss others. 

we optimize the structure of the HF HF complex. The following table lists the results for our AMI, PM3 and HF/6-31+G(d) optimizations as well as an MP2/ 6-31 l-F-tG(2d,2p) tight-convergence optimization taken from the Gaussian Quantum Chemistry Archive  

Basis set keywords are not used for semi-empirical methods as they are inherent in the method s definition. 

Exploring Chemistry with Electronic Structure Methods 113 

The effect of a-n coupling in a series of donor/acceptor organic molecules including zwitterions and betaines has been investigated within an INDO/S sum over state approximation by Rao and Bhanuprakash.177 Yuan et al.m have investigated structural effects on ft using the CNDO/S Cl sum over states approach. 

Theoretical calculations of a CFC substitute, CHC12CF3, have been carried out by Cabral180 using DFT. The calculated polarizability agrees well with an 

Choi et a/.182 have used a perturbed Hartree-Fock method with the PM3 Hamiltonian to analyse the dynamic a, ft and response functions of thiophene, furan, pyrrole, 1,2,4-triazole, 1,3,4-oxadiazole and 1,2,4-thiadiazole monomers and oligomers. The PM3 method is also the basis of a study of the static a and response functions of tetrakis(phenylethynyl)ethene.183 

Barlow et a/.182 have developed a simple orbital model for the interpretation of the quadratic response of metallocene based chromophores of the form metallocene-(7r-bridge)-acceptor. The orbitals have been obtained by DFT. Vance et /.185 have used a simple two state model to analyse the first hyperpolarizability of cyanide bridged structures containing metallic ions. 

The importance of helical structures for enhancing the first hyperpolarizability has been discussed by Panda and Chandrasekhar 193 and SOS theory has been used by Moreau et a/.194 and Monshi et al 95 to investigate excited state polarizations with solvent effects. Torrens et al.196 have developed a scheme to predict molecular polarizabilities from the effect of interacting dipoles. 

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Two review papers that introduce and compare the myriad of semi-empirical methods ... 

Thiel W 1996 Perspectives on semiempirical molecular orbital theory New Methods in Computationai Quantum Meohanios (Adv. Chem. Phys. XCiti) ed I Prigogine I and S A Rice (New York Wiley) pp 703-57 Earlier texts dealing with semi-empirical methods include ... 

The first point to remark is that methods that are to be incorporated in MD, and thus require frequent updates, must be both accurate and efficient. It is likely that only semi-empirical and density functional (DFT) methods are suitable for embedding. Semi-empirical methods include MO (molecular orbital) [90] and valence-bond methods [89], both being dependent on suitable parametrizations that can be validated by high-level ab initio QM. The quality of DFT has improved recently by refinements of the exchange density functional to such an extent that its accuracy rivals that of the best ab initio calculations [91]. DFT is quite suitable for embedding into a classical environment [92]. Therefore DFT is expected to have the best potential for future incorporation in embedded QM/MD. 

This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

Covers theory and applications of ah initio quantum mechanics calculations. The discussions are useful for understanding the differences between ah initio and semi-empirical methods. Although both sections are valuable, the discussion of the applications oi ah initio theory fills a void. It includes comparisons between experiment and many types and levels of calculation. The material is helpful in determining strategies for, and the validity of. ah initio calculations. 

A textbook describing the theory associated with calculation s of Ih e electronic structure of molecti lar system s. While the book focuses on ab ini/rci calculation s, much of the in formation is also relevant to semi-empirical methods. The sections on the Hartree-fock an d Con figuration ItUeracTion s tn elh ods, in particular, apply to HyperChem. fhe self-paced exercisesare useful for the beginning computational chemist. 

Calculated transition structures may be very sensitive Lo the level of theory employed. Semi-empirical methods, since they are parametrized for energy miriimnm structures, may be less appropriate for transition state searching than ab initio methods are. Transition structures are norm ally characterized by weak partial" bonds, that is, being broken or formed. In these cases UHF calculations arc necessary, and sometimes even the inclusion of electron correlation effects. 

Parameters for elements (basis liinctions in ah miiw methods usually derived from experimental data and empirical parameters in semi-empirical methods nsually obtained from empirical data or ah initu> calcii la lion s) are in depen den t of th e ch em -leal environment, [n contrast, parameters used in molecular mechanics methods often depend on the chem ical en viron-ment. 

I hcre arc two types of Cl calculations im piemen ted in Hyper-Ch ern sin gly exciled Cl an d in icroslate Cl. I hc sin gly excited C which is available for both ah initio and sem i-etn pirical calculations may be used to generate CV spectra and the microstate Cl available only for the semi-empirical methods in HyperChern is used to improve the wave function and energies including the electron ic correlation. On ly sin gle point calculation s can he perform cd in HyperChetn using Cl. 

WFth all semi-empirical methods, IlyperChem can also perform psendo-RIfF calculations for open -shell systems. For a doublet stale, all electrons except one are paired. The electron is formally divided into isvo "half electron s" with paired spins. Each halfelec-... 

The five semi-empirical methods in TlyperChem differ in many technical details. Treatm eii i of electron-electron in leraction s is one ma or dislin gnish m g featnre. Anoth er imporlaii 1 dislingnish-ing feature is the approach used to parameterize the methods. Based on the methods used for obtaining parameters, the XDO methods fall into two classes ... 

For small molecules, the accuracy of solutions to the Schrtidinger equation competes with the accuracy of experimental results. However, these accurate a i initw calculations require enormous com putation an d are on ly suitable for the molecular system s with small or medium size. Ah initio calculations for very large molecules are beyond the realm of current computers, so HyperChern also supports sern i-em p irical quantum meclian ics m eth ods. Sem i-em pirical approximate solutions are appropriate and allow extensive cliem ical exploration, Th e in accuracy of the approxirn ation s made in semi-empirical methods is offset to a degree by recourse to experimental data in defining the parameters of the method. 

Dbvioiisly, the ah inmo method in IlyperCi hein is suitable for the former and the semi-empirical methods are more appropriate for the latter. 

IlyperChcm semi-empirical methods usually let you request a calculation on the lowest energy stale of a given multiplicity or the next lowest state of a given spin m ultipliriiy. Sin ce m osl m olecu les with an even num her of electron s are closed-shell singlets without... 

Th c second sum mation of th c above is over tli c orbitals of atom. A, HyperClicrn calculates th c electrostatic potential using this last expression for the semi-empirical methods. 

As with the other semi-empirical methods that we have considered so far, the overlap niJtrix is equal to the identity matrix. The following simple matrix equation must then be solved ... 

Table 1.7 Comparison of quantities calculated with various semi-empirical methods.

Stewart J J P 1989a. Optimisation of Parameters for Semi-empirical Methods 1. Method. Journal of Computational Chemistry 10 209-220. 

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