Methods of molecular quantum mechanics; an introduction to electronic molecular structure

Methods of Molecular Quantum MechanicsAn Introduction to Electronic Molecular Structure Valerio MagnascoUniversity of Genoa, Genoa, Italy... Methods of Molecular Quantum MechanicsAn Intr

Methods of Molecular Quantum Mechanics

An Introduction to Electronic

Molecular Structure

Valerio MagnascoUniversity of Genoa, Genoa, Italy

Methods of Molecular Quantum Mechanics

Methods of Molecular Quantum Mechanics

An Introduction to Electronic

Molecular Structure

Valerio MagnascoUniversity of Genoa, Genoa, Italy

Ó 2009 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Magnasco, Valerio.

Methods of molecular quantum mechanics : an introduction to electronic molecular structure / Valerio Magnasco.

p cm.

Includes bibliographical references and index.

ISBN 978-0-470-68442-9 (cloth) – ISBN 978-0-470-68441-2 (pbk : alk paper) 1 Quantum chemistry.

2 Molecular structure 3 Electrons I Title.

Set in 10.5/13pt, Sabon by Thomson Digital, Noida, India.

Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall.

To my Quantum Chemistry students

1.2.15 Transformations in Coordinate Space 9

1.3.1 Correspondence between Physical Observables

1.3.2 State Function and Average Value

1.3.3 Time Evolution of the State Function 161.4 Physical Interpretation of the Basic Principles 17

3.1 Atomic Orbitals as a Basis for Molecular Calculations 31

3.2.1 Choice of an Appropriate Coordinate System 32

3.2.3 Solution of the Angular Equation 373.2.4 Some Properties of the Hydrogen-like Atomic

4.2.3 The First Excited 2p State of the Hydrogenic

4.2.4 The Ground State of the He-like System 61

4.4.1 The First 1s2s Excited State of the He-like Atom 674.4.2 The First 1s2p State of the He-like Atom 69

5.2 The Pauli Equations for One-electron Spin 78

6 Antisymmetry of Many-electron Wavefunctions 856.1 Antisymmetry Requirement and the Pauli Principle 85

6.3 Distribution Functions 896.3.1 One- and Two-electron Distribution

7 Self-consistent-field Calculations and Model Hamiltonians 997.1 Elements of Hartree–Fock Theory for Closed Shells 100

7.2 Roothaan Formulation of the LCAO–MO–SCF

9.4 Valence Bond Theory and the Chemical Bond 1539.4.1 Schematization of Valence Bond Theory 1539.4.2 Schematization of Molecular Orbital Theory 1549.4.3 Advantages of the Valence Bond Method 1549.4.4 Disadvantages of the Valence Bond Method 1549.4.5 Construction of Valence Bond Structures 156

9.5 Hybridization and Molecular Structure 162

11.1 The H–H Nonexpanded Interactions

12.2 Group Theoretical Methods 252

12.2.3 Representations and Characters 25512.2.4 Three Theorems on Irreducible

The book consists of 12 chapters Particular emphasis is devoted to theRayleigh variational method, the essential tool for any practical applica-tion both in molecular orbital and valence bond theory, and to thestationary Rayleigh–Schroedinger perturbation methods, much attentionbeing given to the Hylleraas variational approximations, which areessential for studying second-order electric properties of molecules andmolecular interactions, as well as magnetic properties In the last chapter,elements on molecular symmetry and group theoretical techniques arebriefly presented Major features of the book are: (i) the consistent usefrom the very beginning of the system of atomic units (au), essential forsimplifying all mathematical formulae; (ii) the introductory use of densitymatrix techniques for interpreting the properties of many-body systems so

as to simplify calculations involving many-electron wavefunctions; (iii) anintroduction to valence bond methods, with an explanation of the origin

of the chemical bond; and (iv) a unified presentation of basic elements ofatomic and molecular interactions, with particular emphasis on thepractical use of second-order calculation techniques Though many ex-amples are treated in depth in this book, for other problems and theirdetailed solutions the reader may refer to the previous book by the author.The book is completed by alphabetically ordered bibliographical refer-ences, and by author and subject indices.

Finally, I wish to thank my son Mario for preparing the drawings at thecomputer, and my friends and colleagues Deryk W Davies and MicheleBattezzati for their careful reading of the manuscript and useful discus-sions In saying that, I regret that, during the preparation of this book,DWD died on 27 February 2008

I acknowledge support by the Italian Ministry for Education Universityand Research (MIUR), under grant number 2006 03 0944 003, and AracneEditrice (Rome) for the 2008 publishing of what is essentially the Italianversion entitled Elementi di Meccanica Quantistica Molecolare

Valerio MagnascoGenoa, 15 May 2009

Principles

The great majority of the applications of molecular quantum mechanics tochemistry are based on what is called the orbital model The planetarymodel of the atom can be traced back to Rutherford (Born, 1962) Itconsists of a point-like nucleus carrying the whole mass and the wholepositive charge þZe surrounded by N electrons each having the elemen-tary negative charge e and a mass about 2000 times smaller than that ofthe proton and moving in a space which is essentially that of the atom.1Electrons are point-like elementary particles whose negative charge isdistributed in space in the form of a charge cloud, with the probability offinding the electron at point r in space being given by

jcðrÞj2

dr¼ probability of finding in dr the electron in state cðrÞ

ð1:1ÞThe functions c(r) are called atomic orbitals (AOs, one centre) ormolecular orbitals (MOs, many centres) and describe the quantum states

of the electron For (1.1) to be true,c(r) must be a regular (or Q-class)mathematical function (single valued, continuous with its first derivatives,

quadratically integrable) satisfying the normalization condition

It seems appropriate at this point first to introduce in an elementary way theessential mathematical methods which are needed in the applications, fol-lowed by a simple axiomatic formulation of the basic postulates of quantummechanics and, finally, by their physical interpretation (Margenau, 1961)

In what follows we shall be concerned only with regular functions of thegeneral variable x

1.2.1 Dirac Notation

Complex conjugate cðxÞ ¼ hcj Y bra



ð1:3Þ

The scalar product (see the analogy between regular functions andcomplex vectors of infinite dimensions) ofcbyc can then be written inthe bra-ket (‘bracket’) form:

then we say that the functionc(x) (the ket jci) is normalized to A (the norm

ofc) The function c can then be normalized to 1 by multiplying it by thenormalization factor N ¼ A1=2

2 In an atom or molecule, there must be zero probability of finding an electron infinitely far from its nucleus.

hcjwi ¼ Nhc0jw0 Sc0i ¼ NðS  SÞ ¼ 0 ð1:9Þ1.2.4 Set of Orthonormal Functions

Let

fwkðxÞg ¼ ðw1w2 wk wi .Þ ð1:10Þ

be a set of functions If

hwkjwii ¼ dki k; i ¼ 1; 2; ð1:11Þwheredkiis the Kronecker delta (1 if i ¼ k, 0 if i 6¼ k), then the set is said to

so that the algebraic sum of two operators is commutative.

In general, the product of two operators is not commutative:

n different independent eigenfunctions belong to it We shall see later thatthe Schroedinger equation for the amplitudec(x) is a typical eigenvalueequation, where ^A ¼ ^H ¼ ^T þ V is the total energy operator (theHamiltonian), ^T being the kinetic energy operator and V the potentialenergy characterizing the system (a scalar quantity).

1.2.10 Hermitian Operators

A Hermitian operator is a linear operator satisfying the so-called over rule’:

‘turn-hcj^Awi ¼ h^Acjwið

The Hermitian operators have the following properties:

(i) real eigenvalues;

(ii) orthogonal (or anyway orthogonalizable) eigenfunctions;(iii) their eigenfunctions form a complete set

Completeness also includes the eigenfunctions belonging to the uous part of the eigenvalue spectrum

contin-Hermitian operators are i@=@x, ir, @2=@x2,r2, ^T ¼ ðh2=2mÞr2and H ¼ ^^ T þ V, where i is the imaginary unit (i2¼ 1),

r ¼ ið@=@xÞ þ jð@=@yÞ þ kð@=@zÞ is the gradient vector operator,

r2¼ r  r ¼ @2=@x2þ @2=@y2þ @2=@z2 is the Laplacian operator(in Cartesian coordinates), ^T is the kinetic energy operator for a particle

of mass m with h ¼ h=2p the reduced Planck constant and ^H is theHamiltonian operator

1.2.13 From Operators to Matrices

Using the expansion theorem we can pass from operators (acting onfunctions) to matrices (acting on vectors; Chapter 2) Consider a finite

3 A less stringent stipulation of completeness involves the approximation in the mean (Margenau, 1961).

n-dimensional set of basis functions fwkðxÞgk ¼ 1; ; n Then, if ^A is aHermitian operator:

1CC

A ¼ w†^Aw ð1:27Þ

which is called the matrix representative of the operator ^A in the basis

fwkg, and we use matrix multiplication rules (Chapter 2) In this way,the eigenvalue equations of quantum mechanics transform into eigen-value equations for the corresponding representative matrices Wemust recall, however, that a complete set implies matrices of infiniteorder

Under a unitary transformation U of the basis functions w ¼

ðw1w2 wnÞ:

the representative A of the operator ^A is changed into

A0¼ w0†^Aw0¼ U†AU ð1:29Þ1.2.14 Properties of the Operator r

We have seen that in Cartesian coordinates the vector operatorr (thegradient, a vector whose components are operators) is defined as(Rutherford, 1962)

r ¼ i @

@xþ j @@yþ k @@z ð1:30Þ

Now, let F(x,y,z) be a scalar function of the space point P(r) Then:

rF ¼ i@F

@xþ j @F@yþ k @F@z ð1:31Þ

is a vector, the gradient of F

If F is a vector of components Fx, Fy, Fz, we then have for the scalarproduct

is the Laplacian operator

From the vector product ofr by the vector F we obtain a new vector, thecurl or rotation of F (written curl F or rot F):

¼ curl F ¼ i curlxFþ j curlyFþ k curlzF ð1:34Þ

a vector operator with components:

curlxF¼ @Fz

@y  @F

y

@zcurlyF¼ @Fx

@z  @F

z

@xcurlzF¼ @Fy

½^Lx; ^Ly ¼ i^Lz; ½^Ly; ^Lz ¼ i^Lx; ½^Lz; ^Lx ¼ i^Ly

1.2.15 Transformations in Coordinate Space

We now give the definitions of the main coordinate systems useful inquantum chemistry calculations (Cartesian, spherical, spheroidal), therelations between Cartesian and spherical or spheroidal coordinates, andthe expressions of the volume element dr and of the operatorsr and r2

inthe new coordinate systems We make reference to Figures 1.1 and 1.2

(i) Cartesian coordinates (x,y,z):

4 Note that the ladder operators are non-Hermitian.

x ¼ r sin u cos w; y ¼ r sin u sin w; z ¼ r cos u ð1:45Þ

Figure 1.2 Cartesian and spheroidal coordinate systems

Figure 1.1 Cartesian and spherical coordinate systems

dr¼ r2dr sin u du dw ð1:46Þ

r ¼ er @

@rþ eu

1r

35

¼ h2 @2

@u2 þ cot u @

@uþ

1sin2u

(iii) Spheroidal coordinatesðm; n; wÞ:

We now formulate in an axiomatic way the basis of quantum mechanics inthe form of three postulates

1.3.1 Correspondence between Physical Obervables

and Hermitian Operators

In coordinate space, we have the basic correspondences

r¼ ix þ jy þ kz )^r ¼ r

p¼ ipxþ jpyþ kpz)^p ¼ ihr



ð1:61Þwhere i is the imaginary unit (i2 ¼  1) and h ¼ h=2p is the reducedPlanck constant More complex observables can be treated by repeated

applications of the correspondences (1.61) under the constraint that theresulting quantum mechanical operators must be Hermitian.5 Kineticenergy and Hamiltonian (total energy operator) for a particle of mass m inthe potential V are examples already seen We now give a few furtherexamples by specifying the nature of the potential energy V.

(a) The one-dimensional harmonic oscillator

If m is the mass of the oscillator of force constant k, then theHamiltonian is

^

H ¼  h2

2mr2þ kx2

(b) The atomic one-electron problem (the hydrogen-like system)

If r is the distance of the electron of mass m and charge e from anucleus of charge þZe (Z ¼ 1 will give the hydrogen atom), then theHamiltonian in SI units6is

^

H ¼  h22mr2 1

The quantities observable in physical experiments must be real.

6 An SI dimensional analysis of the two terms of Equation (1.63) shows that they have the dimension of energy (Mohr and Taylor, 2003): jh 2 r 2 =2mj ¼ ðkg m 2 s 1Þ 2

m 2kg 1¼

kg m 2 s 2¼ J; jZe 2 =4p« 0 rj ¼ C2ðJ C  2 mÞ m  1 ¼ J.

7 Atomic units were first introduced by Hartree (1928a).

At the end of a calculation in atomic units, as we always shall do, theactual SI values can be obtained by taking into account the SI equiva-lents (1.65).

The Hamiltonian of the hydrogenic system in atomic units will thentake the following simplified form:

(c) The atomic two-electron system

Two electrons are attracted by a nucleus of charge þZ TheHamiltonian will be

^h ¼  12r2 Z

is the one-electron Hamiltonian (which has the same functionalform for both electrons) and the last term is the Coulomb repulsionbetween the electrons (a two-electron operator) Z ¼ 2 gives the Heatom

(d) The hydrogen molecule-ion Hþ2

This is a diatomic one-electron molecular system, where the electron issimultaneously attracted by the two protons at A and B The Born–Oppenheimer Hamiltonian (see Chapter 9) will be

(e) The hydrogen molecule H2

This is a diatomic two-electron molecular system The Born–Oppenheimer Hamiltonian will be

^

H ¼ ^h1þ ^h2þ 1

r12þ 1R

is the interatomic potential between A and B

1.3.2 State Function and Average Value of Observables

We assume there is a state function (or wavefunction, in general complex)Y(x,t) that describes in a probabilistic way the dynamical state of amicroscopic system In coordinate space, Y is a regular function ofcoordinate x and time t such that

where integration covers the whole space

The average value of any physical observable8A described by theHermitian operator ^A is obtained from

where integration is extended over all space and ^A acts always on Y andnot onY The last expression above shows that ^A is weighted with the(normalized) probability densityYY.

1.3.3 Time Evolution of the State Function

The state function Y is obtained by solving the time-dependentSchroedinger equation:

^HYðx; tÞ ¼ ih@Yðx; tÞ

a partial differential equation which is second order in the spacecoordinate x and first order in the time t This equation involves theHamiltonian of the system ^H, so that the total energy E is seen to play

a fundamental role among all physical observables

If the Hamiltonian ^H does not depend explicitly on t (the case ofstationary states), then, following the usual mathematical techniques, thevariables in Equation (1.76) can be separated by writingY as the product

of a space functioncðxÞ and a time function g(t):

giving upon substitution

^HcðxÞ ¼ EcðxÞgðtÞ ¼ g0expðivtÞ



ð1:78Þ

where E is the separation constant, g0 is an integration constant, and

v ¼ E=h The first part of Equation (1.78) is the eigenvalue equation forthe total energy operator (the Hamiltonian) of the system, and cðxÞ iscalled the amplitude function This is the Schroedinger equation that wemust solve or approximate for the physical description of our systems Thesecond equation gives the time dependence of the stationary state, whilegeneral time dependence is fundamental in spectroscopy It is immediatelyevident that, for the stationary state, the probabilityYYdx is indepen-dent of time:

Yðx; tÞYðx; tÞ dx ¼ jYðx; tÞj2

dx ¼ jcðxÞj2jg0j2

1.4 PHYSICAL INTERPRETATION OF THE BASIC PRINCIPLES

The explanation of our, so far, unusual postulates is hidden in the nature

of the experimental measurements in atomic physics The experimentallyobserved atomicity of matter (electrons and protons, carrying the ele-mentary negative and positive charge respectively), energy (hn, Planck),linear momentum (h/l, De Broglie), and angular momentum (h, Bohr)implies some limits in the measurements done at the microscopic level.The direct consequence of the ineliminable interaction between theexperimental apparatus and the object of measurement at the subatomicscale was shown by Heisenberg (1930) in his gedanken Experimente, and

is embodied in his famous uncertainty principle, which for canonicallyconjugate quantities9can be stated in the form

where h is the Planck constant, Dx is the uncertainty in the measurement ofthe x-coordinate, and Dpxis the uncertainty resulting in the simultaneousmeasurement of the conjugate linear momentum It can be seen that thecorresponding quantum mechanical operators do not commute:

½x; ^px ¼ x^px ^pxx ¼ ih ð1:81Þwhereas operators that are not conjugate commute; say:

½x; ^py ¼ x^py ^pyx ¼ 0 ð1:82ÞThis means that we cannot measure with the same arbitrary accuracytwo conjugated dynamical variables whose quantum mechanicaloperators do not commute: the exact determination of the positioncoordinate of the electron would imply the simultaneous infinite in-accuracy in the determination of the corresponding component of thelinear momentum!

As a consequence of Heisenberg’s principle, the only possible tion of the dynamical state of a microscopic body is a probabilistic one,and the problem is now to find the function that describes such aprobability This was achieved by Schroedinger (1926a, 1926b, 1926c,1926d), who assumed that particle matter (the electron) could be

descrip-9 In the sense of analytical mechanics.

described by the progressive wave in complex form:

Y ¼ A expðiaÞ ¼ A exp½2piðkx  ntÞ ð1:83Þwhere A is the amplitude and a the phase of a monochromatic plane wave

of wavenumber k and frequency n, which propagates along x Taking intoaccount the relations of De Broglie and Planck connecting wave-like andparticle-like behaviour:

the time-dependent Schroedinger Equation (1.76) giving the time evolution

of the state functionY Hence, the two correspondences (1.87) and (1.88),connecting linear momentum and total energy to the first derivatives ofthe functionY,necessarilyimplythefundamentalrelationsoccurringinouraxiomatic proposition of quantum mechanics

As a consequence of our probabilistic description, in doing experiments

in atomic physics we usually obtain a distribution of the observableeigenvalues, unless the state function Y coincides at time t with theeigenfunctionwkof the corresponding quantum mechanical operator, inwhich case we have a 100% probability of observing for A the value Ak.Such probability distributions fluctuate in time for all observables but theenergy, where we have a distribution of eigenvalues (the possible values of

the energy levels) constant in time These different cases are qualitativelydepicted in the plots of Figure 1.3.

The formulation of quantum mechanics of Section 1.3 is the most usefulfor us, but not the only one possible For instance, using Dirac’s approach,Troup (1968) showed the equivalence between our Schroedinger10‘wavemechanics’ (the continuous approach) and Heisenberg ‘matrix mechan-ics’ (the discrete approach), giving working examples in both cases (e.g.the harmonic oscillator and the hydrogen atom) However, this matter isoutside the scope of the present book

Figure 1.3 Probability distribution of the physical observable A Top row: definite value for the kth eigenvalue Bottom row: fluctuation in time of the probability distribution (left) and distribution of energy eigenvalues constant in time (right)

10

Based on the Schroedinger (1926a, 1926b, 1926c, 1926d) series of four papers entitled

‘Quantisierung als Eigenwertproblem’ The equivalence between matrix and wave mechanics was examined by Schroedinger (1926e) in an intermediate paper.

Matrices

Matrices are the powerful algorithm connecting the differential equations

of quantum mechanics to equations governed by the linear algebra

of matrices and their transformations After a short introduction onelementary properties of matrices and determinants (Margenau andMurphy, 1956; Aitken, 1958; Hohn, 1964), we introduce special matricesand the matrix eigenvalue problem

A matrix A of order m
n is an array of numbers or functions orderedaccording to m rows and n columns:

1C

iAii, the sum of all diagonal elements

Methods of Molecular Quantum Mechanics: An Introduction to Electronic Molecular Structure Valerio Magnasco

Ó 2009 John Wiley & Sons, Ltd

Two matrices A and B of the same order are equal if

B ¼ A Bij ¼ Aijfor all i; j ð2:3ÞMatrices can be added (or subtracted) if they have the same order:

A  B ¼ C Cij¼ Aij Bij ð2:4ÞAddition and subtraction enjoy commutative and associativeproperties

Multiplying a matrix A by a complex number c implies multiplication ofall elements of A by that number:

The product, rows by columns, of two (or more) matrices A by B ispossible if the matrices are conformable (the number of columns of Aequals the number of rows of B):

Xp b¼1

then matrices A and B commute

The product of more than two matrices enjoys the associative property:

The trace of a product of matrices is invariant under the cyclicpermutation of its factors

2.2 PROPERTIES OF DETERMINANTS

Given a determinant |A| of order n (n rows and n columns), we call |Aij| theminor of |A|, a determinant of order ðn  1Þ obtained from A by deletingrow i and column j, and by jaijj ¼ ð  1Þi þ jjAijj the cofactor (signedminor) of |A|

The main properties of determinants will be briefly recalled here(Aitken, 1958)

1 A determinant can be expanded in an elementary way in terms of any

of its rows or columns:

2 A determinant changes sign if we interchange any two rows orcolumns

3 A determinant is unchanged upon interchanging all rows andcolumns

4 The product of |A| by a complex number c implies that only one row(or column) is multiplied by that number

5 A determinant vanishes if two rows or two columns are identical

6 A determinant vanishes if all elements of a row or column are zero

7 If each element in any one row (or column) is the sum of two or morequantities, then the determinant can be written as the sum of two ormore determinants of the same order (distribution property) Forinstance: ja þ bcj ¼ jacj þ jbcj, where for short we write only thediagonal elements

8 The determinant of the product of two or more square matrices ofthe same order is the product of the determinants of the individualmatrices:

jABCj ¼ jAj  jBj  jCj ð2:11Þ