Theoretical Background
A basic understanding of how approximate molecular wave equations are constructed and solved is essential to the proper use of quantum chemical software. Quantum chemistry is highly mathematical in nature, and the language used to describe quantum chemical methods more often relates to equations than to chemical concepts. Non-specialists who are interested in using quantum chemical methods as molecular modeling tools can be faced with a considerable learning curve.
This section is designed to serve as a starting point for acquiring a working knowledge of the current generation of quantum chemical methods. The emphasis in this work is on the relationships between important quantum chemical concepts and their mathematical foundations. This presentation is loosely based on Lowe.
Standing Waves in a Clamped String
The properties of waves in a string clamped at both ends (clamped string) are analogous to some of the important basic quantum mechanical properties of atoms and molecules. Waves can be generated by plucking (adding energy to) such a string:
By observation, waves in a clamped string must adopt discrete wavelengths because the ends of the string cannot be displaced.
Such waves can be mathematically described by solving the wave equation for . The one-dimensional form of the wave equation is sufficient to describe the clamped string model:
The solutions to the wave equation, , are called the wavefunctions. is proportional to the energy density at any point along the wave (i.e. the amount of energy from the pluck that is stored in each part of the string).
All solutions to the wave equation must be symmetric. However, there are two possible kinds of wave symmetries. The first type of symmetry is such that the wave and its reflection about the wave coordinate axis are superimposable (symmetric):
The second type of symmetry is such that the wave and its reflection are not superimposable (antisymmetric):
An asymmetric waveform is shown below for comparison purposes: Wave-Like Particles (the de Broglie Wavelength) The idea that particles have wave-like properties was developed by de Broglie. Planck had shown in 1900 that electromagnetic radiation was emitted and absorbed in discrete quanta having energy proportional to the frequency of the radiation . Einstein showed in 1905 that the energy of a particle is , where m is the particle's relativistic mass. Combining these two expressions yields the relationship between a photon's momentum and frequency (or wavelength):
That is, electromagnetic radiation has particle-like characteristics (momentum) in addition to its wave-like characteristics (wavelength, diffraction). In 1924 de Broglie suggested that matter also had this dual nature and proposed that a wavelength can be associated with the momentum of any particle, not just photons: As a particle's momentum becomes large (i.e. due to its mass), becomes undetectably small. It is useful to express in terms of energy:
Schrodinger's Equation for a One-Electron Atom The connection between classical waves and de Broglie's particle waves was made by Schrodinger. The classical three-dimensional wave equation is:
where is the Laplacian operator () and is the wavefunction describing the displacement at any point along the wave. Schrodinger substituted the de Broglie wavelength for to adapt the classical wave equation to particle waves. The Schrodinger equation is given by:
This equation can be rearranged in a series of algebraic steps to a more convenient form:
H is called the Hamiltonian operator. Substituting H into Schrodinger's equation leads to: is called an eigenfunction and E an eigenvalue. The Hamiltonian Operator The electrostatic potential energy, V, of one charged particle in the field of another is given by:
where q is the charge on each particle and r is the distance between them. If q1 and q2 are both + or both -, they repel (V > 0). If q1 and q2 are opposite, they attract (V < 0). The potential energy between an electron (q = -e) and a nucleus (q = +Ze) can be written as: Substituting for V in the wave equation gives:
V decreases as the electron and nucleus get closer. So, the electron moves faster when it's close to the nucleus, and slower when it's farther away (because energy is interconverted from T into V to maintain constant E). The second derivative of is the rate of change of slope of at any given point, which describes the curvature of "wiggliness" of the function. So, the wigglier is at a given point in space, the greater is the kinetic energy of the electron at that point.
Properties of and E Each and its corresponding energy relate to a single electron bound to a nucleus of charge +Ze. The are called one-electron orbitals (often called hydrogen-like atomic orbitals). They are designated as 1s, 2s, 2p, etc.
The one-electron atomic orbitals are obtained by analytically solving Schrodinger's equation. gives the probability density of finding an electron at a given point in space. Each can be contoured (i.e. graphed at a constant value). Note that each solution has a radial term containing r, and some have angular terms containing and . The s orbitals are spherical, and therefore have no angular dependence. All other orbitals have spatial directionality, and the corresponding wavefunctions have angular terms.
Classically, an electron and nucleus are bound by an attractive potential that varies with their distance, r. T is never sufficiently large for an electron to escape this potential. Thus, V defines a lower bound to E (i.e. E > V for all r).
Under quantum mechanics, however, electrons can exist at larger distances than allowed by V (i.e. E < V). tapers off exponentially outside of this range (i.e. contains an imaginary component). This is called "tunneling". and all chemical properties derived therefrom are real valued, and so is given by , where is the complex conjugate of .
The sum of the probability density over the entire volume of space must reflect the existence of the electron somewhere in space (normalization condition):
where dv is an infinitesimal chunk of volume and the c's are normalization constants that adjust the to satisfy the normalization equation. can be symmetric or antisymmetric, but never unsymmetric. The superposition of all wavefunctions of an atom is spherically symmetric (i.e. there is no favored direction in an atom). Each one-electron wavefunction can exist in two forms ( and ) called "spin" states. Individual wavefunctions on the same atom do not overlap:
The solutions for E are discrete (quantized), having n = 1,2,3... possible values, reminiscent of the n = 1,2,3... wavelengths (given by 2L/n) of a clamped string. Schrodinger's Equation for a Multi-Electron Atom The Hamiltonian for an atom with k electrons is given by:
The Schrodinger equation for a multi-electron atom can be solved numerically, although Velectron-electron cannot be included as an explicit term in the Hamiltonian. Rather, its effect on can be accounted for by a mathematically simpler approach: that each electron interacts with an average of the nucleus + all other electrons (self-consistent field approximation). Neglect of Electron-Electron Repulsion and Separability of the Hamiltonian Consider a two-electron atom (e.g. He) in which Velectron-electron is neglected: where h(i) are one-electron Hamiltonians. Schrodinger's equation can then be approximated as:
where and are the one-electron energies such that (the total energy) and is composed of a combination of one-electron wavefunctions, . Since electron-electron interactions are neglected in this approach, the probability of finding either electron at a given position (i.e. ) does not depend on the position of the other electron. This can be mathematically represented using the products of the one-electron wavefunctions:
The arguments of the wavefunctions denote the positions of each electron. The multi-electron wavefunction must take into consideration the fact that electrons are indistinguishable, and therefore interchanging electron position assignments in a wavefunction cannot lead to a different wavefunction. This is not the case for the products themselves:
but is true of the sum of permuted products: and their difference: where and are the one-electron wavefunctions (e.g. 1s, 2s, etc.). Interchanging electrons in the two equations leads to: and Note that the sign changes for when electron position assignments are interchanged, but the equations are otherwise unaffected by this operation. and are called "space" functions because they depend on the spatial positions of the electrons. Spin Electrons can exist in two possible states called "spin" states designated as and . Spin states are mathematically represented by "spin" functions, and , analogous to and . Spin functions must be symmetric or antisymmetric with respect to the interchange of electron state assignments. For a two electron system, the possible spin functions are: A complete wavefunction, , is composed of a space function ( or ) and a spin function ( or ): In practice, only the antisymmetric form, , is physically meaningful. results from symmetrically opposite constituent space and spin functions: and There are four possible antisymmetric atomic spin orbitals for a 1s(1) 2s(2) configuration:
where the are approximate wavefunctions for the entire atom, made by "mixing" one-electron spin orbitals. Antisymmetric spin orbitals are often constructed using a mathematical function called a Slater determinant. Electron Correlation Effects Velectron-electron was neglected in the Hamiltonian in the previous sections. The effects of electron-electron interactions are, in general, called "electron correlation". and E cannot be used to correctly predict atomic properties without somehow accounting for electron correlation. Multi-electron wavefunctions () are overly influenced by nuclear-electron attraction, and this tends to spatially contract the electron density distribution toward the nucleus. This is contrary to the unaccounted effect of electron-electron repulsion, which tends to make the orbitals larger and more diffuse. The overestimated degree of contraction can be corrected by implicitly reducing the effects of nuclear-electron attraction. The rationale is that each electron is screened from the nucleus by the other electrons. The screening effect is assumed to be averaged over all other electrons. The one-electron wavefunctions () can be improved in this regard by modifying the nuclear charge constant, Z:
can be written as: where is an adjustable parameter. As becomes smaller, the orbital tends to expand. can be determined using the self-consistent field approach (SCF). In practice, more useful forms of are available (e.g. Slater and Gaussian-type orbitals). The Schrodinger Equation for a Molecule The Hamiltonian for a molecule with N atoms and k electrons is given by:
Note that Tnucleus is omitted. This (Born-Oppenheimer) approximation considers the nuclei to be stationary relative to the electron motions. Since the positions of the nuclei are fixed during the calculations, Vnucleus-nucleus is typically treated as a constant. Just as for multi-electron atoms, approximate molecular wavefunctions (molecular spin orbitals) can be created using a set of one-electron wavefunctions. The major differences between the various molecular orbital calculation methods pertains to their consideration (or lack of consideration) of electron correlation. Basis Sets The set of one-electron wavefunctions used to build molecular orbital wavefunctions is called the basis set. The hydrogen-like wavefunctions modified for electron correlation are generally not used per se because they lead to mathematical complications and time-consuming calculations. Instead, wavefunctions are used in which the radial terms, , are simplified. The most common such wavefunctions are Slater-type orbitals (STO):
where s is a screening constant, and Z - s can be taken as and Gaussian-type orbitals (GTO): where is a curve-fitting constant used to approximate an STO. The STO screening constants are calculated for small model molecules using rigorous self-consistent field methods, and then generalized for use with actual molecules of interest. This data is supplied with the various software implementations that use STOs. The mathematical requirements for solving the integrals of the wave equation using STOs are time consuming. The accuracy of STOs can be improved by combining two or more STOs (i.e. with two different values of ) into a single one-electron wavefunction (double- basis set):
Likewise, STOs can be devised that reflect the shape properties of polarized one-electron orbitals (e.g. the combination of a 1s and a 2pz orbital). Gaussian orbitals are mathematically simpler than STOs, but less accurate. A 1s GTO and the corresponding one-electron hydrogen-like orbital are compared in the following plot:
All of the one-electron orbitals can be built by combining sets of gaussian functions (gaussian primitives) that approximate each STO. The result is called a contracted gaussian function. A minimal basis set is one in which only occupied orbitals of each isolated atom are used to compose the molecular orbitals. Unoccupied molecular orbitals are called virtual orbitals. Additional information on basis sets is available (Jan Labanowski, Ohio Supercomputer Center). A library of basis sets is maintained by the Environmental Molecular Sciences Laboratory (Pacific Northwest Laboratory, Battelle Memorial Institute).
My early life was spent in Burnham-on-Sea, Somerset, a small seaside resort town (population around 5000) on the west coast of England. I was born on October 31, 1925 and lived there with my parents until shortly after the end of the Second World War in 1946. No member of my family was involved in any scientific or technical activity. Indeed, I was the first to attend a university.
My father, Keith Pople, owned the principal men's clothing store in Burnham. In addition to selling clothes in the shop, he used to drive around the surrounding countryside with a car full of clothes for people in remote farms and villages. He was resourceful and made a fair income, considering the economic difficulties during the depression of the 1930s. My great-grandfather had come to Burnham around 1850 and set up a number of local businesses. He had a large family and these were split up among his children. As a result, I had relatives in many of the other businesses in the town. My grandfather inherited the clothing shop and this passed to my father when he returned from the army at end of the First World War.
My mother, Mary Jones, came from a farming background. Her father had moved from Shropshire as a young man and had farmed near Bath for most of his life. I suspect that he would have preferred to be a teacher, for he had a large collection of books and encyclopedias. He wanted my mother to be a schoolteacher, but this did not happen. Instead, she became a tutor to children in a rich family and, later, a librarian in the army during the first war. Most of her relatives were farmers in various parts of Somerset and Wiltshire so, as small children, my younger brother and I spent much time staying on farms.
Both of my parents were ambitious for their children; from an early age I was told that I was expected to do more than continue to run a small business in this small town. Education was important and seen as a way of moving forward. However, difficulties arose in the choice of school. There was a good preparatory school in Burnham but, as part of the complex English class system, it was not open to children of retail tradesmen, even if they could afford the fees. The available alternative was unsatisfactory and my parents must have agonized over what to do. Eventually, they decided to send us to Bristol Grammar School (BGS) in the nearest big city thirty miles away. BGS was the prime day school for boys, catering mainly to middle class families resident in the city, although it received a government grant for accepting about thirty boys a year from the state elementary schools. I went there in the spring of 1936 at the age of ten. Some arrangement had to be made for boarding and I used to return home by train each weekend. This I found unappealing and eventually I persuaded my parents to allow me to commute daily - two miles by bicycle, twenty-five miles by train and one mile on foot. I continued to do this during the early part of the war, a challenging experience during the many air attacks on Bristol. Often, we had to wend our way past burning buildings and around unexploded bombs on the way to school in the morning. Many classes had to be held in damp concrete shelters under the playing fields. In spite of all these difficulties, the school staff coped well and I received a superb education.
At the age of twelve, I developed an intense interest in mathematics. On exposure to algebra, I was fascinated by simultaneous equations and rapidly read ahead of the class to the end of the book. I found a discarded textbook on calculus in a wastebasket and read it from cover to cover. Within a year, I was familiar with most of the normal school mathematical curriculum. I even started some research projects, formulating the theory of permutations in response to a challenge about the number of possible batting orders of the eleven players in a cricket team. For a very short time, I thought this to be original work but was mortified to find n! described in a textbook. I then attempted to extend n! to fractional numbers by various interpolation schemes. Despite a lot of effort, this project was ultimately unsuccessful; I was angry with myself when I learned of Euler's solution some years later. However, these early experiences were valuable in formulating an attitude of persistence in research.
All this mathematical activity was kept secret. My parents did not comprehend what I was doing and, in class, I often introduced deliberate errors in my exercises to avoid giving an impression of being too clever. My grades outside of mathematics and science were undistinguished so I usually ended up several places down in the monthly class order. This all changed suddenly three years later when the new senior mathematics teacher, R.C. Lyness, decided to challenge the class with an unusually difficult test. I succumbed to temptation and turned in a perfect paper, with multiple solutions to many of the problems. Shortly afterwards, my parents and I were summoned to a special conference with the headmaster at which it was decided that I should be prepared for a scholarship in mathematics at Cambridge University. During the remaining two years at BGS, I received intense personal coaching from Lyness and the senior physics master, T.A. Morris. Both were outstanding teachers. The school, like many others in Britain, attached great importance to the placement of students at Oxford or Cambridge. Most such awards were in the classics and I think that the mathematics and science staff were very anxious to compete. Ironically, during the last two years at BGS, I abandoned chemistry to concentrate on mathematics and physics. In 1942, I travelled to Cambridge to take the scholarship examination at Trinity College, received an award and entered the university in October 1943.
In the middle of the war, most young men of my age were inducted into the armed forces at the age of seventeen. However, a small group of students in mathematics, science and medicine was permitted to attend university before taking part in wartime research projects such as radar, nuclear explosives, code-breaking and the like. This was a highly successful project and many of my predecessors in earlier years made important contributions to the war effort. The plan was to complete all degree courses in only two years, followed by secondment to a government research establishment. In my case, I completed Part II of the mathematical tripos in May 1945, just as the European war was ending. In fact, it was hard to concentrate on the examinations because of the noisy celebrations going on in the streets outside. The government no longer had need for my services and the university was under great pressure to make room for the deluge of exservicemen as they were demobilized from the armed forces. So, I had to leave Cambridge and take up industrial employment for a period. This was with the Bristol Aeroplane Company, close to where I had attended school. There was little to do there and I had a period of enforced idleness as changing employment was illegal at the time (part of the obsession for a planned economy in postwar Britain).
In 1945, I had little idea of what my future career might be. My interest in pure mathematics began to wane; after toying with several ideas, I finally resolved to use my mathematical skills in some branch of science. The choice of a particular field was postponed, so I devoted much of my time to pestering government offices for permission to return to Cambridge and resume my studies. In the late summer of 1947, I finally received a letter informing me that an unexpectedly large number of students had failed their examinations and a few places were available. So, in October 1947, I returned to Cambridge to begin a career in mathematical science.
Cambridge in 1947 had greatly changed since 1943. The university was crowded with students in their late twenties who had spent many years away at the war. In addition, the lectures were given by the younger generation who had also been away on research projects. There was a general air of excitement as these people turned their attention to new scientific challenges. I remained as a mathematics student but spent the academic year 1947-8 taking courses in as many branches of theoretical science as I could manage. These included quantum mechanics (taught in part by Dirac), fluid dynamics, cosmology and statistical mechanics. Most of the class opted for research in fundamental areas of physics such as quantum electrodynamics which was an active field at the time. I felt that challenging the likes of Einstein and Dirac was overambitious and decided to seek a less crowded (and possibly easier) branch of science. I developed an interest in the theory of liquids, particularly as the statistical mechanics of this phase had received relatively little attention, compared with solids and gases. I approached Fred Hoyle, who was giving the statistical mechanics lectures (following the death of R.H. Fowler). However, his current interests were in the fields of astrophysics and cosmology, which I found rather remote from everyday experience. I next approached Sir John Lennard-Jones (LJ), who had published important papers on a theory of liquids in 1937. He held the chair of theoretical chemistry at Cambridge and was lecturing on molecular orbital theory at the time. When I approached him, he told me that his interests were currently in electronic structure but he would very possibly return to liquid theory at some time. On this basis, we agreed that I would become a research student with him for the following year. Thus, after the examinations in June 1948, I began my career in theoretical chemistry at the beginning of July. I had almost no chemical background, having last taken a chemistry course at BGS at the age of fifteen. Other important events took place in my life at this time. In late 1947, I was attempting to learn to play the piano and rented an instrument for the attic in which I lived in the most remote part of Trinity College. The neighbouring room was occupied by the philosopher Ludwig Wittgenstein, who had retired to live in primitive and undisturbed conditions in the same attic area. There is some evidence that my musical efforts distracted him so much that he left Cambridge shortly thereafter. In the following year, I sought out a professional teacher. The young lady I contacted, Joy Bowers, subsequently became my wife. We were married in Great St. Mary's Church, Cambridge in 1952, after a long courtship. Like many other Laureates, I have benefit immeasurably from the love and support of my wife and children. Life with a scientist who is often changing jobs and is frequently away at meetings and on lecture tours is not easy. Without a secure home base, I could not have made much progress. The next ten years (1948-1958) were spent in Cambridge. I was a research student until 1951, then a research fellow at Trinity College and finally a lecturer on the Mathematics Faculty from 1954 to 1958. Cambridge was an extraordinarily active place during that decade. I was a close observer of the remarkable developments in molecular biology, leading up to the double helix papers of Watson and Crick. At the same time, the X-ray group of Perutz and Kendrew (introduced to the Cavendish Laboratory by Lawrence Bragg) were achieving the first definitive structures of proteins. Elsewhere, Hoyle, Bondi and Gold were arguing their case for a cosmology of continuous creation, ultimately disproved but vigorously presented. Looking through the list of earlier Nobel laureates, I note a large number with whom I became acquainted and with whom I interacted during those years as they passed through Cambridge.
In the theoretical chemistry department, LJ was professor and Frank Boys started as lecturer in September 1948. I began research with some studies of the water molecule, examining the nature of the lone pairs of electrons. This was an initial step towards a theory of hydrogen bonding between water molecules and a preliminary, rather empirical study of the structure of liquid water. This fulfilled my initial objective of dealing with properties of liquids and gained me a Ph.D. and a research fellowship at Trinity College. This highly competitive stage accomplished, I was able to relax a bit and formulate a more general philosophy for future research in chemistry. The general plan of developing mathematical models for simulating a whole chemistry was formulated, at least in principle, some time late in 1952. It is the progress towards those early objectives that is the subject of my Nobel lecture.
At that early date, of course, computational resources were limited to hand calculators and very limited access to motorized electric machines. So my early notes show attempts to simplify theories enough to turn them into practical possibilities. The work paralleling studies of Pariser and Parr led to what became known as PPP theory. This was not a complete model but rather one applicable to systems with only one significant electron per atom. It did fit the general form of conjugated hydrocarbons and achieved some notoriety. In 1953, Bob Parr came to Cambridge to spend a year with Frank Boys. We shared an office and had many valuable discussions; he was to have a major influence on my future. I talked about PPP theory when I began to speak at international meetings in 1955.
In addition to the PPP work, I started theoretical work on other topics in physical chemistry. I began supervision of research students in 1952, beginning with David Buckingham, who completed a masterly thesis on properties of compressed gases. He was the first of a long list of remarkably able and dedicated students who have worked with me over the years. In 1954, LJ was succeeded as professor of theoretical chemistry by Christopher Longuet-Higgins, who was joined by Leslie Orgel shortly afterwards. I continued to spend a lot of time in the chemistry department, although by then I had undertaken new teaching responsibilities as a lecturer in mathematics. The department was crowded and active in those years. Among the many visitors were Linus Pauling, Robert Mulliken, Jack Kirkwood, Clemens Roothaan and Bill Schneider. Frank Boys was also managing a lively group of students.
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