The hydrogenic system, i.e., hydrogen atom or any one-electron-containing-an-ion system, is the only chemical system for which an exact solution of Schrödinger equation is known. In case of the problem related to many electrons atomic systems and molecules, the equations obtained are found to be too complex to be solved exactly. In other words, we can say that apart from a few simple cases, the solution of the Schrödinger equation is impossible in closed form. In many electron systems, the electrons interact with each other and the solutions of the equation obtained become difficult due to many potential energy terms.

Under the above circumstances, approximate methods or theories have been applied to get the solution. These approximate methods or theories are as follows:

• Perturbation theory or method

• Variation theorem or method

We shall acquire knowledge about each theory or method one by one.

Perturbation theory/method for non-degenerate states

If a chemical system is subjected to an electric or a magnetic field, the system becomes disturbed or perturbed. Under this situation, the system under consideration differs from an exactly solvable system by only a small disturbance, enabling an approximation to be carried out by expanding in powers of a smallness parameter.

Let H = perturbed Hamiltonian

H0 = zero order Hamiltonian without disturbance or perturbation of the system or zero order unperturbed Hamiltonian whose solution is known

(H − H0)= change/difference between perturbed and unperturbed Hamiltonian. H differs from H0 by only a small amount.

Then the effect of (H − H0) can be taken into account by utilising the perturbation theory. Further, let us consider that H0 has a spectrum of given values

Then we can express the Schrödinger equation as

It has been assumed that H is slightly different from H0 due to perturbation. Then H can be expressed as a polynomial in a perturbation parameter as

where, H0 = first order perturbation correction

H00 = second order perturbation correction

H000 = third order perturbation correction and so on (the magnitude of

perturbation is found to be inversely proportional to order)

Density functional theory (DFT) is one of the tools of computational chemistry. The density functional calculations are often said to be DFT calculations. It is just like abinitio and semi-empirical calculations based on the Schröodinger equation. It must be kept in mind that DFT does not compute a wave function but it directly gives a method to obtain the electron distribution or electron density function. Therefore, it may be said that the DFT is not based on the wave function, but it is based on the electron density or charge density or electron density function. It is represented or designated by p(x, y, z). The fact is that it is probability per unit volume, which is expressed as p(x, y, z) where P is the probability of finding an electron in a volume element dx . dy . dz. Note that p(x, y, z) is purely a number. It will be clear from the following representation:

The unit of p can be logically expressed as Vol−1. Therefore, p can be mathematically represented by

But for closed shell molecules ni = 2

We have used the words function and functional above. So we shall explain these two mathematical words one by one.

Function

A quantity 0y0 is said to be a function of another quantity 0x0, written as y = f (x), if a change in one produces a change in the other. Thus, in the statement y = 3x2 +5x, i.e., f (x) = 3x2 +5x, y is a function of x and a change in the value of x produces a change in the value of y. If x = 1, then f (1) = 3 ×12 +5 ×1 = 8, if x = 2, then f (2) = 3 ×22 +5 ×2 = 22. Thus, it is clear that the value of ‘y’ is dependent on the value of x.

The function can also be defined in the following manner:

A function f (x) is a mathematical rule, which connects a number with every value of x for which the function f is defined. For example, the function f (x) = x3 +1 connects the number 9 with x = 2, i.e., when we put x = 2, f (2) = 9; similarly, for x = 3 f (3) = (3)2+1 = 28.

The propagation of a periodic disturbance carrying energy is called the wave motion. At any point along the path of the wave motion, a periodic displacement or vibration about a mean position occurs. This may take the form of a displacement of air molecules (e.g., sound waves in air), of water molecules (wave on water), or of elements of a string or wire. The locus of these displacements at any instant is known as wave. The wave motion moves forward a distance equal to its wavelength in the time taken for the displacement at any point to undergo a complete cycle about its mean position. Waves in which the displacement or vibration takes place in the direction of propagation of the waves are called longitudinal waves, e.g., sound waves, but the waves in which the vibration or displacement occurs in a plane at right angles to the direction of propagation of the waves are called transverse waves. We are discussing the waves and wave motion, and therefore, we want to establish a relationship or an equation of wave motion. In order to do so, it will be convenient to consider the simplest type of wave motion, namely, the vibration of string, and then we shall derive an equation of wave motion.

Equation of wave motion

In deriving the equation for wave motion (e.g., vibration of string), let w = the amplitude of vibration at any point, the co-ordinate of which is x at a time t.

we can write

Differentiating Eq. (6.1) with respect to t, we shall get

This is the equation of wave motion in one dimension.

Equation (6.3) is a differential equation and it may be solved by the method of separating the variables, provided u remains constant. Thus w may be written as

w = f (x) .g(t)

where, f (x) = a function of x only

g(t) = a function of time t only

For the motion of standing waves, which occur in a stretched string, it is known to us that g(t) may be expressed as

g(t) = A sin2pƲt

where, Ʋ = frequency of vibration

A = constant, which is maximum amplitude

Therefore, Eq. (6.4) may be expressed as

This Chapter is intended to provide the students of chemistry with the mathematics they need in the course of study of quantum chemistry or quantum mechanics. The students will face difficulties in understanding the theoretical portion of quantum mechanics without the knowledge of mathematics. This chapter will help in developing a strong mathematics background for those who will continue into the mathematics of advanced theoretical chemistry. It is to be noticed that the subject of differential equations, for example, is no longer a series of trick solutions of abstract but the solutions and general properties of the differential equations, the students will most frequently encounter in the description of our real chemical world. The other topics such as matrices, determinants, vectors, etc., are equally important in quantum mechanics.

Differential equations

An equation involving the dependent variable and independent variable and also the derivatives of the dependent variable is called a differential equation.

Ordinary differential equation

A differential equation comprising derivatives with respect to a single independent variable is known as an ordinary differential equation.

For example:

Equations (3.1), (3.2), (3.3) and (3.4) are the examples of ordinary differential equations.

Partial differential equation

A differential equation consisting of partial derivatives with respect to more than one independent variables is known as a partial differential equation.

Order and degree of a differential equation

Order

Order of a differential equation is defined as the order of the highest derivative in the differential equation.

Degree

It is defined as the degree of the highest derivative which occurs in it, after the differential equation has been made free from fractions and radicals as far as the derivative is concerned.

Linear and non-linear differential equation

A differential equation is called linear if (a) every dependent variable and every derivative involved occur in the first degree only and (b) no products of dependent variables and/or derivatives occur.

General solution, particular solution, and arbitrary constants

General solution:

A solution of a differential equation having independent arbitrary constant equal in number to the order of the differential equation is known as its general solution.

Physical Constants

This book is based on the latest syllabi of Indian universities and it is intended for honours and postgraduate students of chemistry, physics and engineering. It begins with black-body radiation and ends with a chapter that discusses Hohenberg–Kohn theorem, Kohn–Sham energy and Kohn–Sham equations along with local density approximation and its applications. Density functional theory is a tool of computational chemistry useful for calculations of atomic and molecular properties. Two special topics (Chapters 3 and 4) are added in the book to provide mathematical foundation of quantum mechanics and quantum chemistry. All important topics are comprehensively covered and explained in a lucid manner so that students can obtain an overall concept of Quantum Chemistry. Solved problems are rightly given in the end of each chapter for self-assessment. Some special questions on concept would help students gain ideas about the related topics. These questions will also help them take preparations for various competitive examinations.

We owe special thanks to Bharat Ratna Professor C.N.R. Rao for kindly writing the foreword for this book. We also express our sincere thanks to Padma Shri Professor J.A.K. Tareen, Vice-Chancellor BSAR University and Dr A.A. Bhalekar, Emeritus Professor of Chemistry, Nagpur University, for their constant suggestions while writing the book. Our thanks are also due to Dr K.K. Nag, former Vice- Chancellor of T.M. Bhagalpur University, Ranchi University and V.B. University for his affection and encouragement.

The insightful discussions the authors had with Dr Awadh Prasad, Head, P.G. Department of Physics, SKMU, Dr A.K. Ghosh, Head, Department of Chemistry, Patna University, Dr R.K. Mehra, University Professor of Chemistry, Commerce P.G. College, Patna should also be sincerely acknowledged here. Their suggestions were a great source of inspiration for us. We are also grateful to Manish Choudhary and Sana Bhanot, both editors at Cambridge University Press, India, for their support and enthusiasm to improve this book at every stage of publication.

We are aware of the fact that a molecule is an aggregate of atomic nuclei encircled by electrons. The situation is this that the nuclei repel each other but are attracted by electrons, which also repel one another. The aggregate molecule will remain stable only if the sum of the attractive forces between nuclei and electrons just balance the repulsive forces among nuclei and among electrons present.

It is to be noted that we have derived the Roothaan's equations and Hartree−Fock self-consistent field (SCF) methods in Chapter 11 under multi-electron system. In this chapter, we shall discuss the matrix form of the Roothaan's equations and also deliberate on the Roothaan's method for quantum mechanical calculations in one dimension.

In this chapter, we shall also deal with the shapes of molecules. For diatomicmolecules, the only structural element is bond length, but in case of polyatomic molecules, both bond lengths and bond angles are important elements, which determine the energy of the molecule. We have studied earlier the nature of bonding in diatomic molecules. We want to extend the discussion of bonding to molecules in which one central atom is joined to other atoms more than one in number. We have discussed while giving idea of the valance bond theory that electrons in a molecule occupy atomic orbitals and in the course of the bond formation, these orbitals overlap. We also learnt that greater the overlapping, stronger will be the bond formation, and the direction of the bond is estimated by the direction in which the two orbitals overlap as far as practicable. Overlapping of atomic orbitals actually yields a stable bond, provided the atomic orbitals have the same energy and adequate symmetry.

In this chapter, we shall also discuss the shape of molecules for which hybridisation will be dealt with. We shall give the quantum mechanical picture of hybridisation of different kinds.

Matrix form of the Roothaan's equations

The Roothaan's equations (Eq. (11.176), Chapter 11) are represented by

In this equation, the coefficients Cja are related to the MOs fa and the basis function cj by the following equation:

In Chapter 1, we discussed about the black-body radiation where in we learnt regarding Wien's law, Rayleigh−Jeans law and Planck's radiation law. Planck's quantum hypothesis or Planck's radiation equation is general by which Wien's Law and Rayleigh−Jeans radiation law can be arrived at. Actually, Wien's law and Rayleigh−Jeans law are special cases of Planck's radiation law. We also studied the photoelectric effect, Compton effect and atomic spectra in the first chapter. The photoelectric effect and Compton effect are vividly indicative of the corpuscular (particle) nature of visible light; X-radiation, interference and diffraction phenomena are evidence of its wave nature. Therefore, it can be concluded that the motion of photons clearly obeys special laws in which corpuscular and wave characteristics are combined. In the same chapter, we dealt with the atomic spectrum of hydrogen. Analysis of hydrogen atom spectrum should convince us that light is emitted or absorbed in energy quanta, i.e., in particles, which we know as photons. It does not mean that we must revive the old controversy of the corpuscular versus the undulatory theories of light. On the contrary, we know now that both theories are part of a more general theory. In some instances one theory applies and in others we must invoke the alternative.

Einstein explained the photoelectric effect by Planck's quantum theory and he also handled the specific heat of solids considering the said theory. In the present chapter, we shall focus on wave−particle duality and Heisenberg's uncertainty principle and their verification by experiments.

Dual nature of electron/de Broglie wave

Keeping in view that light has dual nature, in 1924 de Broglie suggested that a small particle like electron would also behave both as a particle and as a wave. This is called the dual nature electron or matter wave or de Broglie wave.

We know that the energy of a photon can be written in terms of Planck's quantum theory, i.e.,

E = hʋ

But according to Einstein theory of relativity energy can be expressed as

E = mcs2

where, m = mass of electron; c = velocity of light; h = Planck's constant; ʋ = frequency of radiation.

Useful Integrals

The basic principle of Hǜckel molecular orbital (HMO) theory lies in the fact that it is absolutely necessary a one-electron treatment and the electrons remain in the p orbitals. In other words, we can assume that both the core electrons and the electrons present in the skeleton are regarded as “frozen”. It is pertinent to note that there is no interaction between s and p electrons present in the molecule.

Hǜckel (as early as in 1931) pointed out that it was possible to state the characteristics of conjugated hydrocarbons and polyenes by the quantum mechanical model, which took into consideration only p electrons. The Hǜckel HMO approximation works best for a class of alternant hydrocarbons (may be aliphatic or aromatic).

The explicit starting point for the derivation of Hǜckel method for p electron system is the Eigen value formulation of Schröodinger equation, HΨ = EΨ. Hǜckel applied this equation to molecules, keeping in view that H and Y represent molecular Hamiltonian and wave function, respectively.

Multiplying HΨ = EΨ by Ψ we have

Also multiplying both the sides by the volume element dt, Eq. (13.1) takes the form

Integrating and rearranging the above equation, we can express this as

This equation represents the energy value/expectation value of energy of the system.

Next, the molecular wave function will be approximated as a linear combination of atomic orbitals (LCAO) by using suitable basis function fi. Keep in mind that the combination of n basis function will give rise to n molecular orbitals, which can be expressed as

where, Ci = variational coefficients

and Fi = basis functions

Substituting the value of Ψ in Eq. (13.2), we have Substituting these in the above equation, we shall obtain where, N = numerator

D = denominator

Differentiating Eq. (13.5) with respect to Ci and on applying the condition of minimum to get the minimum value of E, we can write

where, N0 = first differentiation of N

D0 = first differentiation of D

The above equation can also be expressed as

From this, we shall get the following set of linear equations:

These linear equations can be expressed in a more general form as

When one proceeds to solve Eq. (13.8), one will find for each root the ratio of the expansion coefficients, and finally, one will get the coefficients uniquely by normalising each orbital.

Before taking into consideration the quantum mechanical operators, it is perhaps the best to clear the meaning of the term operator in the mathematical sense. The term operator may be defined in the following manner:

It means, operator. function = new function

Here symbol d/dx is an operator that alters the function sin x into first derivative with respect to x, i.e., cos x.

Similarly, the symbol () is an operator, the instruction being to take the square root of the quantity, which follows it. For example, if a number, say 25, is kept under operator (), it alters 25 into its square root, i.e., 25 = 5. Addition, subtraction, multiplication, division, log, differentiation, integration, etc., are also operators. It is to be noted that the operators always operate on the function written to the right of them and the operator has no physical significance if written alone.

An operator is normally written with a cap sign (ˆ) overhead, e.g., Â. Thus,  represents an operator. To make it clearer, if we write  f (x) = g(x), it means  is an operator operating on the function f (x). g(x) is an another function, which is a new one obtained after the operation of operator  on the function f (x).

If  = d/dx and f (x) = ax2, then  f (x) = d/dx(ax2) = 2ax = g(x). Similarly, we can cite other examples similar to the above.

In the eighteenth century, it was believed that light consisted of small particles as earlier postulated by a famous scientist Newton in his corpuscular theory. Various experimental studies were carried out later, which supported the wave theory of light. In the later part of the nineteenth century, a large number of experimental facts/observations came to light, which could not be explained by wave theory. One of the important experimental studies was performed by various scientists on black-body radiation, which led to the development of quantum theory. The pioneers among the scientists in the field of black-body radiation were Stefan−Boltzmann, Rayleigh−Jeans, Wein and Max Planck. A brief discussion on black-body radiation and distribution of energy in temperature radiation is presented in the subsequent sections.

Black-body radiation

A black-body is defined as a perfect absorber and emitter of radiation. Actually, it absorbs all the radiations falling on it, without reflecting any. Experiments have shown that all hot bodies when perfectly blackened, gives rise to a thermal spectrum which depends on the temperature only, and not on the chemical composition or the material the body is made of.

In practice, black-body radiation is studied by making a cavity, which is insulated so that the only energy that can be absorbed is the energy supplied in the form of heat to enhance the temperature of the cavity. The cavity is evacuated and a small hole is made on one side of the apparatus through which the radiation can pass. One approximation suggests that the radiation coming from the hole in the cavity becomes a better sample radiation inside the cavity. In short, we can say that a black-body, therefore, comprises the following:

• It must emit radiation of the same character and intensity as it absorbs.

• It must emit radiation of nearly all wavelengths, and at a particular temperature

• It must absorb radiations of all wavelengths completely.

The intensity radiation emitted from the hole is then studied as a function of wavelength at various temperatures of the cavity. On the basis of the experimental results of the black-body radiation, a graph between energy distribution El andwavelength l is plotted at various temperatures. It is shown in Fig. 1.1.