An Overview of Computational Chemistry

Chem. 141

Dr. Mack

Theoretical Chemistry: The mathematical description of chemistry Computational Chemistry: A mathematical method that is sufficiently well developed that it can be automated for implementation on a computer.

Chemical Problems Computer Programs

Physical Models Math formulas

1 Note that the words exact and perfect do not appear in these definitions.

•Computational chemistry is based on a approximations and assumptions.

•Only a real experimental measurement can approach the limits of exactness!

What does Computational Chemistry Calculate? Energy, Structure, and Properties • Molecular Geometries: • What is the energy for a given geometry? • How does energy vary when geometry changes? • Which geometries are stable? • How does energy change w/r extenal perturbation?

2 What else can be computed: •Enthalpies of formation •Dipole moment •Orbital energy levels (HOMO, LUMO, others) •Ionization energy (HOMO energy) •Electron affinity (LUMO energy) •Electron distribution (electron density) •Electrostatic potential •Vibrational frequencies and normal modes (IR spectra) •Electronic excitation energy (UV-Vis spectra) •NMR chemical shifts and coupling constants •Reaction path and barrier height •Reaction rate

What can one learn from computational calculations?

• Molecular Visualization (Graphic Representation) • Molecular Mechanics (Classical Newtonian Physics) • Semi-empirical Molecular Orbital Theory Quantum • Ab Initio Molecular Orbital Theory Mechanical • Density Functional Theory Methods • Geometry Optimization • Molecular Dynamics

3 Types of Calculations:

Molecular Mechanics (MM) Empirical energy functions parameterized against experimental dataFast, simple, generally not very accurate (> 104 atoms) Semi-empirical Molecular Orbital (MO) Methods Can treat moderate sized molecules (> 102 atoms). Accuracy depends on parameterization. Ab-initio Molecular Orbital Methods Computationally demanding (> 10 atoms) I will focus on Accuracy can be systematically improved. these topics Density Function Theory (DFT) More efficient than ab-initio calculations (> 10 atoms) Accuracy varies, however, there is no systematical way to improve the accuracy.

Types of Computational Calculations:

Ab Initio: The term "Ab Initio" is latin for "from the beginning".

•Computations of this type are derived directly from theoretical principles, with no inclusion of experimental data. •Mathematical approximations are usually a simple functional form for a an approximate solution to a differential equation. A wave function! (The Shrödinger Equation) The most common type of ab initio calculation is called a Hartree-Fock calculation. The ? of complicated system (molecule) is generated from a linear combination of simple functions (basis set). The parameters for the SE are varied until the solution (energy of the system) is optimized.

4 Molecular Transition States properties Reaction coords.

Ab initio electronic structure theory Hartree-Fock (HF) Spectroscopic Electron Correlation (MP2, CI, CC, etc.) observables

Geometry Prodding prediction Benchmarks for Experimentalists parameterization

Goal: Insight into chemical phenomena. 9

Setting up the problem…

What is a molecule? A molecule is “composed” of atoms, or, more generally as a collection of charged particles, positive nuclei and negative electrons.

The interaction between charged particles is described by; Coulomb Potential

qj rij qiq j qiq j Vij = V (rij) = = 4pe0rij rij qi

Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena.

5 But, electrons and nuclei are in constant motion… In Classical Mechanics, the dynamics of a system (i.e. how the system evolves in time) is described by Newton’s 2nd Law: F = force F = ma a = acceleration dV d2r r = position vector - = m 2 dr dt m = particle mass

In Quantum Mechanics, particle behavior is described in terms of a wavefunction, Y. ¶Y Hˆ Y = ih Time-dependent Schrödinger Equation t ¶ i = -1;h = h 2p ( ) Hˆ Hamiltonian Operator

Time-Independent Schrödinger Equation

Hˆ (r,t) = Hˆ (r) If H is time-independent, the time- -iEt / h dependence of Y may be separated out Y (r,t) = Y(r)e as a simple phase factor.

Hˆ (r)Y(r) = EY(r) Time-Independent Schrödinger Equation

Describes the particle-wave duality of electrons.

6 Hamiltonian for a system with N-particles

Hˆ = Tˆ + Vˆ Sum of kinetic (T) and potential (V) energy N N h2 N h2 æ ¶ 2 ¶ 2 ¶ 2 ö Tˆ = Tˆ = - Ñ 2 = - ç + + ÷ Kinetic energy å i å 2m i å2m ¶x 2 ¶y 2 ¶z2 i=1 i=1 i i=1 i è i i i ø

æ 2 2 2 ö 2 ¶ ¶ ¶ Ñi = ç 2 + 2 + 2 ÷ Laplacian operator è ¶xi ¶yi ¶zi ø

N N N N q q ˆ i j V = ååVij = åå Potential energy r i=1 j>1 i=1 j>1 ij (Coulombic) When these expressions are used in the time-independent Schrodinger Equation, the dynamics of all electrons and nuclei in a molecule or atom are taken into account. 13

Born-Oppenheimer Approximation

• Since nuclei are much heavier than electrons, their velocities are much smaller. To a good approximation, the Schrödinger equation can be separated into two parts: – One part describes the electronic wave function for a fixed nuclear geometry. – The second describes the nuclear wave function, where the electronic energy plays the role of a potential energy. + ˆ ˆ ˆ ˆ ˆ ˆ H = Tn + Te + Vne + Vee +Vnn

n = nuclear e = electronic ne = nucleus-electron ee = electron-electron nn = nucleus-nucleus

7 • Since the motions of the electrons and nuclei are on different time scales, the kinetic energy of the nuclei can be treated separately. This is the Born-Oppenheimer approximation. As a result, the electronic wave function depends only on the positions of the nuclei.

• Physically, this implies that the nuclei move on a potential energy surface (PES), which are solutions to the electronic Schrödinger equation. Under the BO approx., the PES is independent of the nuclear masses; that is, it is the same for isotopic molecules.

E H. + H. 0

H H • Solution of the nuclear wave function leads to physically meaningful quantities such as molecular vibrations and rotations.

Limitations of the Born-Oppenheimer approximation

•The total wave function is limited to one electronic surface, i.e. a particular electronic state. •The BO approx. is usually very good, but breaks down when two (or more) electronic states are close in energy at particular nuclear geometries. •In such situations, a “ non-adiabatic” wave function - a product of nuclear and electronic wave functions - must be used.

The electronic Hamiltonian becomes, ˆ ˆ ˆ ˆ ˆ H = Te +Vne + + Vee +Vnn

B.O. approx.; fixed nuclear coordinates

8 Hartree-Fock Self-consistent Field (SCF) Theory

GOAL: Solve the electronic Schrödinger equation, HeY=E Y.

PROBLEM: + – Exact solutions can only be found for one-electron systems, e.g., H2 . SOLUTION: – Use the variational principle to generate approximate solutions. Variational principle

– If an approximate wave function is used in He Y =E Y, then the energy must be greater than or equal to the exact energy. – The equality holds when Y is the exact wave function. In practice: – Generate the “best” trial function that has a number of adjustable parameters. – The energy is minimized as a function of these parameters.

The energy is calculated as an expectation value of the Hamiltonian operator:

ˆ Y | H e | Y E = Y | Y If the wave functions are orthogonal and normalized (orthonormal),

dij = 1 Yi | Yj = dij (Kroenecker delta) dij = 0

Then, ˆ E = Y | H e |Y

ˆ ˆ ˆ ˆ ˆ H = Te +Vne + + Vee +Vnn

9 Hartree Approximation • assume that a many electron wave function can be written as a product of one electron functions

Y(r1,r2 ,r3 ,L) = f(r1 )f(r2 )f(r3 )L

• Use the variational method energy, solving the many electron Schrödinger equation is reduced to solving a series of one electron Schrödinger equations • In this approximation, each electron interacts with the average distribution of the other electrons

Variational Theorem

• the expectation value of the Hamiltonian is the variational energy ò Y*Hˆ Ydt = Evar ³ Eexact ò Y*Ydt • the variational energy is an upper bound to the lowest energy of the system • any approximate wave function will yield an energy higher than the ground state energy • parameters in an approximate wave function can be varied to minimize the Evar • this yields a better estimate of the ground state energy and a better approximation to the wave function

10 Since electrons are fermions, S = 1/2, the total electronic wave function must be antisymmetric (change sign) with respect to the interchange of any two electron coordinates. (Pauli principle - no two electrons can have the same set of quantum numbers.) Each electron resides in a spin-orbital, a product of spatial and spin functions.

F(1,2) = f1a(1)f2b(2) - f1a(2)f2b(1)

jI – spin-orbit wave function a = spin up state and b = spin down state (for electrons 1 & 2)

(Spin functions are orthonormal: a | a = b | b =1; a | b = b | a = 0) Interchange the coordinates of the two electrons,

F(2,1) = f1a(2)f2b(1) - f1a(1)f2b(2) F(2,1) = -F(1,2) 21

A more general way to represent antisymmetric electronic wave functions is in the form of a determinant. For the two-electron case,

f1a(1) f2b(1) F(1,2) = = f1a(1)f2b(2) - f1a (2)f2b(1) f1a(2) f2b(2)

For an N-electron N-spinorbital wave function,

f1(1) f2(1) L fN (1) f (2) f (2) L f (2) F = 1 2 N , f |f = d SD L L L L i j ij f N f (N) L f (N) 1( ) 2 N A Slater Determinant (SD) satisfies the antisymmetry requirement. Columns are one-electron wave functions, molecular orbitals. Rows contain the electron coordinates.

One more approximation: The trial wave function will consist of a single SD.

Now the variational principle is used to derive the Hartree-Fock equations... 22

11 Fock Equation

• take the Hartree-Fock wavefunction

Y = f1 f2 L fn • put it into the variational energy expression ò Y*Hˆ Ydt Evar = ò Y*Ydt • minimize the energy with respect to changes in the orbitals

¶Evar / ¶fi = 0 • yields the Fock equation ˆ Ffi = eifi

ˆ Ffi = eifi

• the Fock operator is an effective one electron Hamiltonian for an orbital f • e is the orbital energy • each orbital f sees the average distribution of all the other electrons • finding a many electron wave function is reduced to finding a series of one electron orbitals

ˆ ˆ ˆ ˆ ˆ F = T + VNE + J - K

12 ˆ ˆ ˆ ˆ ˆ F = T + VNE + J - K

- h 2 kinetic energy operator Tˆ = Ñ2 2me

nuclei - e2Z nuclear-electron attraction operator ˆ A Vne = å A riA Coulomb operator (electron-electron repulsion) electrons e2 Jˆf = { f f dt}f i å ò j j i j rij

exchange operator electrons e2 Kˆ f = { f f dt}f (antisymmetry requirement) i å ò j i j j rij

Solving the Fock Equations ˆ Ffi = eifi

1. obtain an initial guess for all the orbitals fi

2. use the current fI to construct a new Fock operator

3. solve the Fock equations for a new set of fI

4. if the new fI are different from the old fI, go back to step 2. 5. When the values converge (do not change to a prescribed limit), the calculation is complete.

ˆ ei = fi '| Fi |fi ' The results are interpreted as MO energies.

13 Hartree-Fock Orbitals

• Hartree-Fock orbitals (f ‘s ) can be computed numerically for atoms • These resemble the shapes of the hydrogenic orbitals (s, p, d and f)

• for homonuclear diatomic molecules, the Hartree-Fock + orbitals resemble the MO’s for species like H2

• s, p, bonding and anti-bonding orbitals 27

LCAO Approximation

• numerical solutions for the Hartree-Fock orbitals only practical for atoms and diatomics • diatomic orbitals resemble linear combinations of atomic orbitals

• e.g. sigma bond in H2 s » 1sA + 1sB • for polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)

f = åcm c m m

14 Basis Functions

f = åcm c m m • c’s are called basis functions • usually centered on atoms • can be more general and more flexible than atomic orbitals • larger number of well chosen basis functions yields more accurate approximations to the molecular orbitals

Basis Set Approximation • For atoms and diatomic molecules, numerical HF methods are available. • In most molecular calculations, the unknown MOs are expressed in terms of a known set of functions - a basis set. Two criteria for selecting basis functions. I) They should be physically meaningful. ii) computation of the integrals should be tractable. • It is common practice to use a linear expansion of Gaussian functions in the MO basis because they are easy to handle computationally. • Each MO is expanded in a set of basis functions centered at the nuclei and are commonly called Atomic Orbitals. (Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).

15 Basis sets • Basis functions approximate orbitals of atoms in molecule • Linear combination of basis functions approximates total electronic wave function • Basis functions are linear combinations of Gaussian functions – Contracted Gaussians – Primitive Gaussians STOs vs. GTOs •Slater-type orbitals (J.C. Slater) •Represent electron density well in valence region and beyond (not so well near nucleus) •Evaluating these integrals is difficult

Gaussian-type orbitals (F. Boys) •Easier to evaluate integrals, but don’t represent electron density well •This can be overcome this by using linear combination of GTOs

Minimal basis set

• One basis function for every atomic orbital required to describe the free atom • Most-common: STO-3G • Linear combination of 3 Gaussian-type orbitals fitted to one Slater-type orbital

• CH4: H(1s); C(1s,2s,2px,2py,2pz)

16 More basis functions per atom

• Split valence basis sets • Double-zeta: • Triple-zeta:

Ways to increase a basis set

• Add more basis functions per atom •allow orbitals to “change size” • Add polarization functions •allow orbitals to “change shape” • Add diffuse functions for electrons with large radial extent • Add high angular momentum functions

17 Ab Initio (latin, “from the beginning”) Quantum Chemistry Summary of approximations • Born-Oppenheimer Approx. • Non-relativistic Hamiltonian • Use of trial functions, MOs, in the variational procedure • Single Slater determinant • Basis set, LCAO-MO approx. Consequence of using a single Slater determinant and the Self-consistent Field equations:

Electron-electron repulsion is included as an average effect. The electron repulsion felt by one electron is an average potential field of all the others, assuming that their spatial distribution is represented by orbitals. This is sometimes referred to as the Mean Field Approximation.

Electron correlation has been neglected!!!

How does one know what to use when?

• Considerations: – Computational resources – Molecule size – Kind of chemical question – Desired accuracy – Recommendations from literature

18 Literature •Books

•F. Jensen, Introduction to Computational Chemistry, Wiley, 1999 (our textbook)

•A. R. Leach, Molecular Modelling, 2nd Ed., Prentice Hall, 2001

•C. J. Cramer, Essentials of Computational Chemistry, 2nd Ed., Wiley, 2004

There are more, but these three should be good at the beginning.