Introduction to Computational Chemistry: Theory

1 Introduction to Computational Chemistry: TheoryDr Andrew GilbertRm 118, Craig Building, Course Lectures - 2011 IntroductionHartree Fock TheoryBasis SetsLecture 11 IntroductionBackgroundThe Wave EquationComputing Chemistry2 Hartree Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory3 Basis SetsBasis functionsAdditional types of functionsComputational AspectsIntroductionHartree Fock TheoryBasis SetsBackgroundComputational chemistryComputational chemistry is the modeling of chemicalphenomenon using computers rather than models used vary in their sophistication:CheminformaticsMolecular mechanicsSemi-empirical methodsAb initioquantum chemistryAll these methods, except the last, rely on empiricalinformation (parameters, energy levelsetc.)

2 In this course we will focus onab initioquantum Fock TheoryBasis SetsBackgroundAb initio quantum chemistryAb initiomeans from the beginning or from firstprinciples , mechanics. The fundamental laws necessary for the mathematical treatment of a large part ofphysics and the whole of chemistry are thus completely known, and the difficulty liesonly in the fact that application of these laws leads to equations that are too complex tobe solved. DiracOver the last four decades powerful molecular modellingtools have been developed which are capable of accuratelypredicting properties of developments have come about largely due to:The dramatic increase in computer design of efficient quantum chemical Fock TheoryBasis SetsBackgroundAdvantagesCalculations aresafe: many experiments have an intrinsic dangerassociated with : there are no waste chemicals effective: compared to performing : many experiments can be difficult to may also be to give greater insight into thechemistry by providing more information about the Fock TheoryBasis SetsBackgroundDisadvantagesCalculations can also beapplied uncritically: just because you can do somethingdoesn t mean you.

3 There may be approximations in the models used,or bugs in the black box: software packages insulate chemists from theunderlying chemistry is not a replacement for experimentalstudies, but plays an important role in enabling chemists to:Explain and rationalise known chemistryExplore new or unknown chemistryIntroductionHartree Fock TheoryBasis SetsThe Wave EquationTheoretical modelThe theoretical foundation for quantum chemistry is thetime-independent Schr odinger wave equation: H =E is the Wavefunction. It is a function of the positions of allthe fundamental particles (electrons and nuclei) in thesystem. His the Hamiltonian operator. It is the operator associatedwith the observable the Total Energy of the system. It is a scalar (number).Relativistic effects are usually small and will be Fock TheoryBasis SetsThe Wave EquationThe HamiltonianThe Hamiltonian, H, is an operator.

4 It contains all the termsthat contribute to the energy of a system: H= T+ V Tis the kinetic energy operator: T= Te+ Tn Te= 12 i 2i Tn= 12MA A 2A 2is the Laplacian given by: 2= 2 x2+ 2 y2+ 2 z2 IntroductionHartree Fock TheoryBasis SetsThe Wave EquationThe Hamiltonian Vis the potential energy operator: V= Vnn+ Vne+ Vee Vnnis the nuclear-nuclear repulsion term: Vnn= A<BZAZB|RA RB| Vneis the nuclear-electron attraction term: Vne= iAZA|RA ri| Veeis the electron-electron repulsion term: Vee= i<j1|ri rj|IntroductionHartree Fock TheoryBasis SetsThe Wave EquationAtomic unitsAll quantum chemical calculations use a special system of unitswhich, while not part of the SI, are very natural and greatlysimplify expressions for various length unit is the bohr (a0= 10 11m)The mass unit is the electron mass (me= 10 31kg)The charge unit is the electron charge (e= 10 19C)The energy unit is the hartree (Eh= 10 18J)For example, the energy of the H atom is 12hartree (exactly).

5 In more familiar units this is 1,313 Fock TheoryBasis SetsThe Wave EquationThe hydrogen atomWe will use the nucleus as the centre of our Hamiltonian is then given by: H= Tn+ Te+ Vnn+ Vne+ Vee= 12 2r 1rThe ground-state wavefunction is simply a function Fock TheoryBasis SetsComputing ChemistryThe chemical connectionSo far we have focused mainly on obtaining the totalenergy of our chemical properties can be obtained from derivativesof the energy with respect to some external of external parameters include:Geometric parameters (bond lengths, anglesetc.).Applied electric fields ( a solvent)Magnetic field (in NMR experiments).1stand 2ndderivatives are commonly available and derivatives are required for some properties, butare expensive (and difficult!) to Fock TheoryBasis SetsComputing ChemistryLecture 1 summaryDifferent models for Chemistry: Cheminformatics (functional groups)Molecular mechanics (atoms)Semi-empirical models (approximate QM)Quantum chemistry (fundamental particles)The Schr odinger wave equation: H =E The Hamiltonian is made up of energy terms.

6 H= Tn+ Te+ Vnn+ Vne+ VeeThe wavefunction gives a complete description of properties are obtained from derivatives of theenergy with respect to external Fock TheoryBasis SetsLecture 21 IntroductionBackgroundThe Wave EquationComputing Chemistry2 Hartree Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory3 Basis SetsBasis functionsAdditional types of functionsComputational AspectsIntroductionHartree Fock TheoryBasis SetsThe molecular orbital approximationHartree-Fock theoryHF Theory is the simplest wavefunction-based relies on the following approximations:The Born-Oppenheimer approximationThe independent electron approximationThe linear combination of atomic orbitals approximationThe Hartree-Fock model introduces an intrinsic error calledthe correlation forms the foundation for more elaborate electronicstructure Fock TheoryBasis SetsThe molecular orbital approximationThe Born-Oppenheimer approximationNuclei are much heavier than electrons (the mass of aproton 2000 times that of an electron) and thereforetravel much more assume the electrons can react instantaneously to anymotion of the nuclei (think of a fly around a rhinoceros).

7 This means the nuclei are stationary the assumption allows us to factorise the wave equation: (R,r) = n(R) e(r;R)where the ; notation indicates a parametric potential energy surface is a direct consequence ofthe BO Fock TheoryBasis SetsThe molecular orbital approximationThe independent electron approximationConsider the H2molecule:The total wavefunction involves 4x3 spatial coordinates: = (R1,R2,r1,r2)We invoke the Born-Oppenheimer approximation: = n(R1,R2) e(r1,r2)How do we model e(r1,r2)?IntroductionHartree Fock TheoryBasis SetsThe molecular orbital approximationThe Hartree wavefunctionWe assume the wavefunction can be written as a Hartreeproduct: (r1,r2) = 1(r1) 2(r2)The individual one-electron wavefunctions, iare calledmolecular form of the wavefunction does not allow forinstantaneous interactions of the , the electrons feel the average field of all the otherelectrons in the Hartree form of the wavefunction is is sometimescalled the independent electron Fock TheoryBasis SetsThe molecular orbital approximationThe Pauli principleOne of the postulates of quantum mechanics is that thetotal wavefunction must be antisymmetric with respect tothe interchange of electron is a consequence of the Pauli Hartree wavefunction is not antisymmetric.

8 (r2,r1) = 1(r2) 2(r1)6= (r1,r2)We can make the wavefunction antisymmetric by adding allsigned permutations: (r1,r2) =1 2[ 1(r1) 2(r2) 1(r2) 2(r1)]IntroductionHartree Fock TheoryBasis SetsThe molecular orbital approximationThe Hartree-Fock wavefunctionThe antisymmetrised Hartree wavefunction is called theHartree-Fock can be written as a Slater determinant: =1 N! 1(r1) 2(r1) N(r1) 1(r2) 2(r2) N(r2).. 1(rN) 2(rN) N(rN) This ensures the electrons are indistinguishable and aretherefore associated with every orbital!A Slater determinant is often written as| 1, 2,.. N IntroductionHartree Fock TheoryBasis SetsThe molecular orbital approximationThe LCAO approximationThe HF wavefunction is antisymmetric and written in termsof the one-electron molecular orbitals (MOs).

9 What do the MOs look like?We write them as a linear combination of atomic orbitals: i(ri) = C i (ri)The are atomic orbitals or basis iare MO Fock TheoryBasis SetsThe molecular orbital approximationAn exampleThe H2molecule: 1= =1 2( A1s+ B1s) 2= =1 2( A1s B1s)For H2the MO coefficients,C i, are 1 2 IntroductionHartree Fock TheoryBasis SetsThe molecular orbital approximationThe HF energyIf is normalised, the expectation value of the energy isgiven by:E= | H| For the HF wavefunction, this can be written:EHF= iHi+12 ij(Jij Kij)Hiinvolves one-electron terms arising from the kineticenergy of the electrons and the nuclear attraction two-electron terms associated with the coulombrepulsion between the two-electron terms associated with theexchange of electronic Fock TheoryBasis SetsThe molecular orbital approximationThe HF energyRemember that our wavefunction is given in terms of adeterminant:| 1, 2.

10 N And our MOs are written as a LCAO: i(ri) = C i (ri)We can write the one-electron parts of the energy as:Hi= i| h| i = C iC i | h| TheJijandKijmatrices can also be written in terms of theMO coefficients,C Fock TheoryBasis SetsThe self-consistent fieldThe variational principleThe MO coefficients,C i, can be determined using thevariational theorem:Variational TheoremThe energy determined from any approximate wavefunction willalways be greater than the energy for the exact energy of the exact wavefunction serves as a lowerbound on the calculated energy and therefore theC icanbe simply adjusted until the total energy of the system isminimised. This is the variational Fock TheoryBasis SetsThe self-consistent fieldThe self-consistent field methodConsider a 2-electron system with MOs 1(r1)and 2(r2).