Computational Chemistry

2018

Lecturer: Dos. Vesa Hänninen

vesa.hanninen@helsinki.fi

Room B432

Contents of the course

Lectures

• Principles of quantum chemistry

• Hartree Fock theory

• Wave function methods

• Density functional methods

• Semiempirical methods

• Graphical models

• Molecular dynamics

Computer exercises

• Introduction to supercomputer environment (CSC)

• Introduction to computation chemistry programs

• Calculations of molecular properties

• Computer simulations of molecular structures, vibrations, chemical reactions,…

• Relies on results of theoretical chemistry and computer science

o Theoretical chemistry is defined as mathematical description of chemistry. It is development of algorithms and computer programs to predict chemical properties.

o Computational chemistry is application of existing computer programs and methodologies to specific chemical problems.

• Practice of efficient computer calculations to obtain chemical and physical properties of molecules, liquids and solids

+ =

What is computational chemistry?

There are two different aspects to computational chemistry:

1. Computational studies can be carried out to find a starting point for a laboratory synthesis, or to assist in understanding experimental data, such as the position and source of spectroscopic peaks.

2. Computational studies can be used to predict the possibility of so far entirely unknown molecules or to explore reaction mechanisms that are not readily studied by experimental means.

Thus, computational chemistry can assist the experimental chemist or it can challenge the experimental chemist to find entirely new chemical objects.

Importance of computational chemistry

Computational methods are considered important in scientific community, see:

The top 100 papers in science

Top cited papers in physics of last 110 years

The Most Cited Chemistry Papers Published, 2004–14

The most cited chemistry articles, 2005

Also, computational chemistry has featured in a number of Nobel Prize awards, most notably in 1998and 2013.

Some applications

Nanotechnological modelling and simulations

Atmospheric science

Drug design

There are many self-sufficient software packages used by computational chemists. Some include many methods covering a wide range, while others concentrating on a very specific range or even a single method. Details of most of them can be found in:

•Biomolecular modelling programs: proteins, nucleic acid.

•Molecular mechanics programs.

•Quantum chemistry and solid state physics software supporting several methods.

•Molecular design software

•Semi-empirical programs.

•Valence bond programs.

Computational chemistry programs

• Theory level: Is the precision high enough for the application?

• The computer programs: Efficiency and implementation

• Computer resources: Is there enough power for the task?

Challenges and limitations

CSC IT Center for Science Ltd is administered by the Ministry of Education, Science and Culture.

CSC maintains and develops the state-owned centralised IT infrastructure and uses it to provide nationwide IT services for research, libraries, archives, museums and culture as well as information, education and research management.

Researchers can use it’s large collection of scientificsoftware and databases.

CSC has offices in Espoo's Keilaniemi and in the RenforsinRanta business park in Kajaani.

About CSC

• Ab initio that use rigorous quantum mechanics

+ accurate

– computationally expensive

good results for small systems ~102 atoms

• Semi empirical that use approximate quantum mechanics

– relies on empirical or ab initio parameters

+ affordable and in some cases accurate

limited to well defined systems with ~104 atoms

• Molecular mechanics that use classical mechanics

– relies on empirical force fields without accounting for electronic properties (no bond breaking or forming)

+ very affordable and used as a virtual experiment

can handle extremely large systems ~109 atoms

Methods of computational chemistry

Principles of quantum chemistry

• The state of the system is specified by a (normalized) wave function 𝜓

• For every measurable property (observable) of a system such as energy 𝐸 for example, there exista corresponding operator (𝐻 for 𝐸)

• Observables satisfy the eigenvalue equation. For example 𝐻𝜓 = 𝐸𝜓.

• The expectation value of the observable, for example 𝐸, is given by

𝐸 = −∞

∞

𝜓∗𝐻𝜓 𝑑𝜏

or using Dirac’s notation as

𝐸 = 𝜓 𝐻 𝜓

Majority of computational chemistry revolves around finding a solution to the static Schrödingerequation

𝐻𝜓 = 𝐸𝜓

The list of closed-form analytic solutions is VERY short. The list of famous chemical problems includes the H atom, the harmonic oscillator, the rigid rotor, the Morse potential, and the ESR/NMR problem.

Hydrogen atomatomic orbitals can be used as a basis for the molecular orbitals.

Harmonic oscillatorbasis for the molecular vibrational motion

Morse oscillatorbasis for the molecular stretching vibration

Rigid rotorbasis for molecular rotational motion

Yields approximate solution for the Schrödinger equation. Variational principle states that the expectation value of the Hamiltonian for trial wavefunction 𝜙 must be greater than or equal to the actual ground state energy

𝜙 𝐻 𝜙

𝜙 𝜙= 𝐸[𝜙] ≥ 𝐸0

Example: Trial function expanded as a linear combination eigenfunctions of the hydrogen ground state 𝜓0

and the first exited state 𝜓1

𝜙 = 𝑐0 𝜓0 + 𝑐1 𝜓1

where 𝑐0 and 𝑐1 are unknown coefficients, so called “variational parameters”. The hydrogenic energy corresponding to this trial function is

𝐸 𝜙 =𝑐0

2 𝜓0 𝐻ℎ 𝜓0 + 𝑐12 𝜓1 𝐻ℎ 𝜓1 + 𝑐0𝑐1 𝜓0 𝐻ℎ 𝜓1 + 𝑐1𝑐0 𝜓1 𝐻ℎ 𝜓0

𝑐02 + 𝑐1

2 =𝑐0

2𝐸0 + 𝑐12𝐸1

𝑐02 + 𝑐1

2 ≥ 𝐸0

Variational method

Any variations in the trial function 𝜙 = 𝑐0 𝜓0 + 𝑐1 𝜓1 which lower the energy expectation value arebringing the approximate energy closer to the exact value.

The best solution can be obtained via optimization, i. e. searching the values of variational parameters𝑐𝑖 which minimize the energy.

𝜕𝐸

𝜕𝑐𝑖= 0

𝑐0

𝑐1

𝐸

For hydrogen atom, the optimal solution is obviously found when 𝑐0 = 1 and 𝑐1 = 0.

In reality we don’t know the eigenfunctions. Instead we use basis functions which are physically relevant for the problem in hand.

𝜙 = 𝑐0 𝜑0 + 𝑐1 𝜑1

The hydrogenic energy corresponding to this trial function is

𝐸 𝜙 =𝑐0

2 𝜑0 𝐻ℎ 𝜑0 + 𝑐12 𝜑1 𝐻ℎ 𝜑1 + 𝑐0𝑐1 𝜑0 𝐻ℎ 𝜑1 + 𝑐1𝑐0 𝜑1 𝐻ℎ 𝜑0

𝑐02 + 𝑐1

2

𝜕𝐸

𝜕𝑐0=

2𝑐0 𝜑0 𝐻ℎ 𝜑0 + 2𝑐1 𝜑0 𝐻ℎ 𝜑1

𝑐02 + 𝑐1

2 −2𝑐0𝐸

𝑐02 + 𝑐1

2 = 0

→ 𝜑0 𝐻ℎ 𝜑0 − 𝐸 𝑐0 + 𝜑0 𝐻ℎ 𝜑1 𝑐1 = 0

𝜕𝐸

𝜕𝑐1= 0 → 𝜑0 𝐻ℎ 𝜑1 𝑐0 + ( 𝜑1 𝐻ℎ 𝜑1 − 𝐸)𝑐1 = 0

We have a system of linear equations. According to linear algebra the energy eigenvalues can beobtained by finding solutions of a characteristic equation

𝜑0 𝐻ℎ 𝜑0 − 𝐸 𝜑0 𝐻ℎ 𝜑1

𝜑0 𝐻ℎ 𝜑1 𝜑1 𝐻ℎ 𝜑1 − 𝐸=

𝐻0,0 − 𝐸 𝐻0,1

𝐻0,1 𝐻1,1 − 𝐸= 0

The energies can be obtained by diagonalizing the Hamiltonian matrix

𝐻 =𝜑0 𝐻ℎ 𝜑0 𝜑0 𝐻ℎ 𝜑1

𝜑0 𝐻ℎ 𝜑1 𝜑1 𝐻ℎ 𝜑1=

𝐻0,0 𝐻0,1

𝐻0,1 𝐻1,1

𝑑𝑖𝑎𝑔 𝐸0 00 𝐸1

In general when we have N basis functions determinant of a characteristic polynomial becomes

𝐻0,0 − 𝐸 ⋯ 𝐻0,𝑁

⋮ ⋱ ⋮𝐻0,𝑁 … 𝐻𝑁,𝑁 − 𝐸

= 0

The secular determinant for N basis functions gives an N-th order polynomial in which is solved for Ndifferent roots, each of which approximates a different eigenvalue.

Example: For a helium atom we can choose the trial function as follows:

𝜙 𝑟1, 𝑟2 = 𝐶 1 + 𝑝𝑟12 + 𝑞 𝑟1 − 𝑟22 exp −𝛼 𝑟1 + 𝑟2

where 𝐶 is normalization constant and 𝑝, 𝑞, and 𝛼 are variational parameters.

After optimization:

𝑝 = 0.30, 𝑞 = 0.13, and 𝛼 = 1.816

𝐸 = −2.9024 a.u. (Three parameters)

𝐸 = −2.9037462 a.u. (1024 parameters)

𝐸 = −2.9037843 a.u. (Experimental value)

Using one parameter trial function 𝜙 = 𝐶∙exp −𝛼 𝑟1 + 𝑟2 the minimumenergy is

𝐸 = −2.848 a.u.

Born-Oppenheimer approximation

Hamilton operator includes the kinetic and potential energy parts of the electrons and nuclei

𝐻 = 𝑇 + 𝑉

The kinetic energy operator for electrons and nuclei can be written as

𝑇 = −ℏ

2

𝑖

𝑛∇𝑖

2

𝑚e−

ℏ

2

𝑗

𝑁∇𝑗

2

𝑚n

where ∇𝑖2=

𝜕2

𝜕𝑥𝑖2 +

𝜕2

𝜕𝑦𝑖2 +

𝜕2

𝜕𝑧𝑖2, 𝑛 and 𝑁 are numbers, and 𝑚e and 𝑚n are masses of electrons and nuclei,

respectively. The potential energy operator includes the electron-electron, nuclei-nuclei, and electron-nucleiparts

𝑉 =1

2

𝑖

𝑛

𝑖′≠𝑖

𝑛𝑒2

4𝜋𝜀0 𝐫𝑖 − 𝐫𝑖′+

1

2

𝑗

𝑁

𝑗′≠𝑗

𝑁𝑍𝑗𝑍𝑗′𝑒2

4𝜋𝜀0 𝐑𝑗 − 𝐑𝑗′−

𝑖

𝑛

𝑗

𝑁𝑍𝑗𝑒

2

4𝜋𝜀0 𝐫𝑖 − 𝐑𝑗

Where 𝐫𝑖 and 𝐑𝑗 are positions of electrons and nuclei and 𝑍𝑗 is the atomic number of nuclei 𝑗. In practice, it is

impossible to solve the Scrödinger equation for the total wavefunction Ψ(𝐫𝑖 , 𝐑𝑗) exactly.

Practical solution: Let’s approximate the wavefunction in a form, where it is factorized in electronicmotion and nuclear motion parts

𝜓(𝐫𝑖 , 𝐑𝑗) ≈ 𝜓el(𝐫𝑖; 𝐑𝑗)𝜓n (𝐑𝑗)

where function 𝜓(𝐫𝑖; 𝐑𝑗) describes electronic motion (depending parametrically on the positions of

nuclei) and function 𝜓(𝐑𝑗) describes the nuclear motions (vibrations and rotations). With this

assumptions, the problem can be reformulated to two separate Scrödinger equations:

𝐻el𝜓el 𝐫𝑖; 𝐑𝑗 = 𝑉(𝐑𝑗)𝜓el 𝐫𝑖; 𝐑𝑗

𝐻n𝜓n 𝐑𝑗 = 𝐸n𝜓n 𝐑𝑗

The former equation is for the electronic problem, considering the nuclei to be fixed. The eigenvalue𝑉(𝐑𝑗) can be called as interatomic potential, which is then used as a potential energy for the latter

equation for the nuclear motion problem.

This procedure, the so called Born-Oppenheimer approximation, is justified because electron is lighter than the proton by the factor 2000, the electron quickly rearranges in response to the slower motion of the nuclei.

Example: For the fixed nuclear positions the H2+-ion the total energy operator can be

written as

We can write the above Hamiltonian further simplified by using atomic units a.u. or Hartree as

𝐻 = −∇2

2−

1

𝑟1−

1

𝑟2+

1

𝑅

The ground state trial wave function is

𝜓 = 𝑐 𝜓1𝑠1+ 𝜓1𝑠2

where the 1𝑠 functions are the type

𝜓1𝑠𝑖=

1

𝜋

1

𝑎0

3 2

𝑒− 𝑟𝑖 𝑎0

Where constant 𝑎0 is the Bohr radius. Thenormalization is

𝑐 =1

2 + 2𝑆12

Where the 𝑆12 is the overlap integralbetween 𝜓1𝑠1

and 𝜓1𝑠2functions

Note that because 𝐑 = 𝐫1 − 𝐫2, the overlap integral and thus the ground electronic state dependsparametrically on the distance between the two nuclei 𝑅

𝑅/bohr

𝑉(𝑅)

The BO–approximation is justified when the energy gap between ground and excited electronic states is larger than the energy scale of the nuclear motion.

The BO–approximation breaks down when

• for example in metals, some semiconductors and graphene the band gab is zero leading to coupling between electronic motion and lattice vibrations (electron-phonon interaction)

• electronic transitions becomes allowed by vibronic coupling (Herzberg-Teller effect)

• ground state degeneracies are removed by lowering the symmetry in non-linear molecules (Jahn-Teller effect)

• interaction of electronic and vibrational angular momenta in linear molecules (Renner-Teller effect)

The Jahn–Teller effect is responsible for the tetragonal distortion of the hexaaquacopper(II) complex ion, [Cu(OH2)6]2+, which might otherwise possess octahedral geometry. The two axial Cu−O distances are 238 pm, whereas the four equatorial Cu−O distances are ~195 pm.

In the vicinity of conical intersections, the Born–Oppenheimer approximation breaks down, allowing non-adiabatic processes to take place. The location and characterization of conical intersections are therefore essential to the understanding to a wide range of reactions, such as photo-reactions, explosion and combustion reactions, etc.

Potential energy surfaces

• The geometry of the molecule is defined using appropriate coordinate system (cartesian coordinates, internal coordinates, Jacobi coordinates, etc.).

• The so called reaction coordinate, which is important in describing the energy profile of a chemical reaction, is simply some combination of bond distances and angles.

• The potential energy surface (PES) is a relationship between energy of a molecular system and its geometry

• The BO-approximation makes the concept of molecular geometry meaningful, makes possible the concept of PES, and simplifies the application of the Scrödinger equation to molecules.

• Among the main tasks of computational chemistry are to determine the structure and energy of molecule and of thetransition states (TS) involved in chemical reactions.

• The positions of the energy minima along the reactioncoordinate give the equilibrium structures of the reactantsand products. Similarly, the position of the energy maximumdefines the transition state.

• Reactants, Products, and transition states are all stationarypoints on the potential energy surface. This means that for system with N atoms all partial derivatives of the energyrespect to each of the 3𝑁 − 6 independent geometricalcoordinates (𝑅𝑖) are zero:

𝜕𝑉

𝜕𝑅𝑖= 0 𝑖 = 1,2, … , 3𝑁 − 6

In the one-dimensional case, or along the reaction coordinate, reactants and products are located in the energy minima and are characterized by a positive second energy derivative

𝑑2𝑉

𝑑𝑅2 > 0

The transition state is characterized by a negative second energy derivative

𝑑2𝑉

𝑑𝑅2 < 0

In the many-dimensional case, each independent geometrical coordinate, 𝑅𝑖, gives rise to 3𝑁 − 6

second derivatives: 𝜕2𝑉

𝜕𝑅𝑖𝑅1,

𝜕2𝑉

𝜕𝑅𝑖𝑅2, … ,

𝜕2𝑉

𝜕𝑅𝑖𝑅3𝑁−6. Thus, it is not possible to say whether any given

coordinate corresponds to a stationary point. To see the correspondence, a new set of coordinates 𝜉𝑖, referred as normal coordinates, is used. They have the property that their cross terms or non-diagonal

terms in second energy derivative matrix vanish: 𝜕2𝑉

𝜕𝜉1𝜉2= 0, etc. For the energy minima:

𝜕2𝑉

𝜕𝜉𝑖2 > 0 𝑖 = 1,2, … , 3𝑁 − 6

Stationary points for which all but one of the second derivatives positive are so-called saddlepoints and may correspond to transition states. If they do, the normal coordinate for which thesecond derivative is negative is referred to as the reaction coordinate 𝜉𝑟:

𝜕2𝑉

𝜕𝜉𝑟2 < 0

The free energy of the system, which drives thechemical reaction, is strictly speaking not the potentialenergy calculated from the electronic Scrödingerequation. As we would expect from Heisenberg’suncertainty principle molecules vibrate incessantly evenin the 0 K. Moreover, in the higher temperatures theentropy also contributes. Thus, when calculatingmeaningful energy differences and thermodynamicproperties the vibrational energy levels are required. Specifically, the zero point vibrational energy (ZPVE) is needed to obtain accurate energy differences for example in the rate constant calculations.

Geometry optimization is the process of starting with an input structure ”quess” and finding a stationary point on the PES. It’susually checked whether the stationary point is a minimum or a transition state by calculating its vibrational frequencies. In transition state one of the vibrations will possess negative (orimaginary) harmonic frequency.

ZPVE corrected potentialenergy difference

Electronic energy

The harmonic vibrational frequency for a diatomicmolecule A-B is

𝜈 =1

2𝜋

𝑘

𝜇

where 𝑘 is the force constant, which is the secondderivative of the potential energy with respect to thebond length 𝑅

𝑘 =𝑑2𝑉 𝑅

𝑑𝑅2

and 𝜇 is the reduced mass

𝜇 =𝑚A𝑚B

𝑚A + 𝑚B

The vibrations of molecules

Polyatomic systems are treated in similar manner. Here, the force constants are second energyderivatives respect to the normal coordinates.

A nonlinear molecule with N atoms has 3N-6 (linear molecule has 3N-5) vibrational degrees of freedom(or vibrational modes). In the harmonic approximation, the potential energy is

𝑉 𝑞1, 𝑞2, … , 𝑞𝑁 =1

2

𝑖=1

𝑁

𝑗=1

𝑁𝜕2𝑉

𝜕𝑞𝑖𝜕𝑞𝑗𝑞𝑖𝑞𝑗

Where 𝑞𝑖 are displacements in some appropriate coordinate system (for example internal coordinatedisplacements). We can find a new set of coordinates (Wilson’s GF method) that simplify the aboveequation to the form:

𝑉 𝜉1, 𝜉2, … , 𝜉𝑁 =1

2

𝑖=1

𝑁𝜕2𝑉

𝜕𝜉𝑖2 𝜉𝑖

2

Where 𝜉𝑖 are the normal coordinates of the molecule. All normal modes are independent in theharmonic approximation.

Vibrations of a methylene group (-CH2-) in a molecule for illustration

Asymmetricalstretching

BendingSymmetricalstretching

Rocking Wagging Twisting

Electron spin

Electron spin is introduced in quantum equations by writing electron wavefunction as a product of spatial wavefunction 𝜓(𝑟) and spin wavefunction. For example H atom 1seigenfunctions take the form

𝜓100

12

𝑟, 𝜎 =1

𝜋

1

𝑎0

3 2

𝑒− 𝑟 𝑎0𝛼

𝜓100−

12

𝑟, 𝜎 =1

𝜋

1

𝑎0

3 2

𝑒− 𝑟 𝑎0𝛽

where 𝜎 is non-spatial spin variable. These two eigenfunction have the same energybecause the total energy operator does not depend on spin. Integration of spin wavefunction over spin variable is defined formally as

𝛼∗𝛼 𝑑𝜎 = 1, 𝛽∗𝛽 𝑑𝜎 = 1, 𝛼∗𝛽𝑑𝜎 = 0, and 𝛽∗𝛼𝑑𝜎 = 0

• Wave functions describing many-electron systems must change sign (be antisymmetric) under theexchange of any two electrons

• We describe n-electron wavefunction using the notation 𝜓 1,2, … , 𝑛 = 𝜓 𝑟1𝜎1, 𝑟2𝜎2, … , 𝑟𝑛𝜎𝑛 . Theposition variables are supressed in favor of keeping track of the electrons.

• The antisymmetric two-electron wave function must satisfy:

𝜓 1,2 = −𝜓 2,1

For example if we write the antisymmetric wavefunction for ground-state of Helium as

𝜓 1,2 = 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 2 𝛽 2 − 𝜓1𝑠 2 𝛼 2 𝜓1𝑠 1 𝛽 1

Clearly we see that if the coordinates of electrons have the same values, the wavefunctionvanishes

𝜓 1,2 = 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 1 𝛽 1 − 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 1 𝛽 1 = 0

This obeys the so called Pauli exclusion principle: Two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously.

For 𝑛-electron system the antisymmetric wave function can be written as Slater determinants and have theform

𝜓 1,2, … , 𝑛 =1

𝑛!

𝜓1 1 𝛼 1 𝜓1 1 𝛽 1 ⋯ 𝜓𝑚 1 𝛽 1

𝜓1 2 𝛼 2 𝜓1 2 𝛽 2 ⋯ 𝜓𝑚 2 𝛽 2⋮ ⋮ ⋱ ⋮

𝜓1 𝑛 𝛼 𝑛 𝜓1 𝑛 𝛽 𝑛 … 𝜓𝑚 𝑛 𝛽 𝑛

where 𝑚 = 𝑛 2 if 𝑛 is even and 𝑚 = 𝑛 + 1 2 if 𝑛 is odd.

• Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. The electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms therefore cannot be squeezed too closely together.

• Electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or Coulombic force.

Hartree-Fock theory

The Hartree method

1928

D.R. Hartree

(1897-1958)

Cambridge, UK

The Hartree-Fock method

1930

V.A. Fock

(1898–1974)

Leningrad, Russia

• In computational chemistry, the Hartree-Fock method has central importance

• If we have the HF solution, the accuracy can systematically be improved by applying various techniques

• It is based on variational approach

• HF is an approximate method, close in spirit to the mean-field approach widely used in solid state and statistical physics

Five major simplifications:

1. The Born-Oppenheimer approximation is inherently assumed.

2. Relativistic effects are completely neglected.

3. Each energy eigenfunction is assumed to be describable by a single Slater determinant.

4. The mean field approximation is implied.

5. The variational solution is assumed to be a linear combination of a finite number of basis functions.

Employing BO-approximation the (non-relativistic) electronic Schrödinger equation can be written as

−

𝑛

∇𝑛2

2−

𝐴

𝑛

𝑍𝐴

𝑅𝐴𝑛+

1

2

𝑛

𝑚≠𝑛

1

𝑟𝑛𝑚𝜓 = 𝐸𝜓

When wavefunction 𝜓 is expressed as a single slater determinant (we approximate the real wavefunctionusing products of single electron wavefunctions) of doubly occupied spatial orbitals we can write theenergy as

𝐸 = 2

𝑖

𝜖𝑖0

+

𝑖

𝑗

2𝐽𝑖𝑗 − 𝐾𝑖𝑗

where the first zeroth-order energy term 𝜖𝑖0

is the energy without electron-electron repulsion, 𝐽𝑖𝑗 is the

Coulomb integral and 𝐾𝑖𝑗 is the exchange integral

𝐽𝑖𝑗 = 𝜓𝑖∗ 1 𝜓𝑗

∗ 21

𝑟12𝜓𝑗 2 𝜓𝑖 1 𝑑𝜏1𝑑𝜏2

𝐾𝑖𝑗 = 𝜓𝑖∗ 1 𝜓𝑗

∗ 21

𝑟12𝜓𝑖 2 𝜓𝑗 1 𝑑𝜏1𝑑𝜏2

We now wish to find the orbitals which lead to a minimum value of the energy. The treatment is simplifiedif we define coulomb and exchange operators as follows

𝐽𝑗𝜓𝑖 1 = 𝜓𝑗∗ 2

1

𝑟12𝜓𝑗 2 𝑑𝜏2 𝜓𝑖 1

𝐾𝑗𝜓𝑖 1 = 𝜓𝑗∗ 2

1

𝑟12𝜓𝑖 2 𝑑𝜏2 𝜓𝑗 1

Note that the exchange operator 𝐾𝑗 exchanges electron 1 and 2 between the two orbitals 𝜓𝑖 and 𝜓𝑗. The

coulomb and exchange integrals can be written as

𝐽𝑖𝑗 = 𝜓𝑖∗ 1 𝐽𝑗𝜓𝑖 1 𝑑𝜏1

𝐾𝑖𝑗 = 𝜓𝑖∗ 1 𝐾𝑗𝜓𝑖 1 𝑑𝜏1

It is seen that the coulomb operator is potential energy which would arise from interaction between

electron 1 and an electron 2 with electron distribution 𝜓𝑗 22

. Such operators represent effective

potentials for an electron moving in the repulsive field of other electrons. The exchange operator has no classical analog, since it arises from the nonclassical antisymmetry principle.

The Hartree-Fock equation for some space orbital 𝜓𝑖 occupied by electron 1 is

ℎ10

+

𝑗

2𝐽𝑗 − 𝐾𝑗 𝜓𝑖 1 = 𝐹 1 𝜓𝑖 1 = 𝜖𝑖𝜓𝑖 1

Where ℎ10

ℎ10

𝜓𝑖 1 = 𝜖𝑖0

𝜓𝑖 1 is the ”hydrogen-like” Hamiltonian operator for electron 1 in the

field of bare nucleus (or nuclei), 𝐹 1 is Fock operator and 𝜖𝑖 is the orbital energy.

These equations show very clearly that in order to solve the one-electron orbital 𝜓𝑖 1 , it is necessary to know wavefunctions 𝜓𝑗 in order to set up the operators 𝐽𝑗 and 𝐾𝑗. The orbitals 𝜓𝑖 1 only account for

the presence of other electrons in an average manner (mean-field theory).

The Hartree–Fock method is also called the self-consistent field method (SCF), meaning that the final field as computed from the charge distribution is required to be "self-consistent" with the assumed initial field.

When the Hartree-Fock equations are solved by numerical integration methods, the procedure in unwieldy. Even worse, the method is incapable of being extended to molecules (molecular orbital theory). The key development (presented by Roothan) was to expand the orbitals 𝜓𝑖 as a linear combination of a set of one-electron basis functions. Introducing a basis set transforms the Hartree-Fock equations into the Roothaanequations. Denoting the atomic orbital basis functions as 𝜙𝑘, we have the expansion

𝜓𝑖 =

𝑘

𝑐𝑘𝑖 𝜙𝑘

This leads to

𝐹

𝑘

𝑐𝑘𝑖 𝜙𝑘 = 𝜖𝑖

𝑘

𝑐𝑘𝑖 𝜙𝑘

Left multiplying by 𝜙𝑙∗ and integrating yields a matrix equation

𝑘

𝑐𝑘𝑖 𝜙𝑙∗𝐹𝜙𝑘𝑑𝜏 =

𝑘

𝑐𝑘𝑖 𝐹𝑘𝑙 = 𝜖𝑖

𝑘

𝑐𝑘𝑖 𝜙𝑙∗𝜙𝑘𝑑𝜏 = 𝜖𝑖

𝑘

𝑐𝑘𝑖 𝑆𝑘𝑙

The Roothaan equations form a set of linear equations in the unknowns 𝑐𝑘𝑖. For a nontrivial solution, wemust have

det 𝐹𝑘𝑙 − 𝜖𝑖𝑆𝑘𝑙 = 0

It is a secular equation which roots give the orbital energies 𝜖𝑖. The matrix equation can be writtenshortly as

𝐅𝐜 = 𝐒𝐜𝜖

Where 𝐅 is the Fock matrix, 𝐒 is the overlap matrix and 𝜖 is a diagonal matrix of the orbital energies 𝜖𝑖.

The Roothan equations must be solved by an iterative process:

One starts with guesses for the orbital expressions as linear combinations of the basis functions 𝜓𝑗. This

initial set of orbitals is used to calculate the Fock operator 𝐹. The matrix elements are calculated, and the secular equation is solved to give an initial set of 𝜖𝑖’s; these 𝜖𝑖’s are used to solve matrix equation for an improved set of coeffifients 𝑐𝑘𝑖, giving an improved orbitals 𝜓𝑖, which are the used to calculate an improved 𝐹, and so on. One keeps going until no improvement in orbital coefficients and energiesoccurs from one cycle to the next.

Example: Helium atom ground state SCF calculation

• Basis set of two 1s STOs:

𝜙𝑘 =1

𝜋

𝜁𝑘

𝑎0

3 2

𝑒− 𝜁𝑘𝑟 𝑎0 , 𝑘 = 1,2 , 𝜁1 = 1.45, 𝜁2 = 2.91

By trial and error, these have been found to be the optimum 𝜁′s to use for this basis set.

• The starting ground state wavefunction: 𝜓𝑗 = 𝑐1𝜙1 + 𝑐2𝜙2 with initial guess: 𝑐1 = 0.6922 and 𝑐2 =

0.3461.

• The Roothan matrix elements

𝐹𝑘𝑙 = 𝜙𝑘 𝐹 𝜙𝑙 = 𝜖𝑘𝑙0

+ 2 𝜙𝑘 𝐽𝑗 𝜙𝑙 − 𝜙𝑘 𝐾𝑗 𝜙𝑙

𝐽𝑗𝜙𝑙 1 = 𝜓𝑗∗ 2

1

𝑟12𝜓𝑗 2 𝑑𝜏2 𝜙𝑙 1

𝐾𝑗𝜙𝑙 1 = 𝜓𝑗∗ 2

1

𝑟12𝜙𝑙 2 𝑑𝜏2 𝜓𝑗 1

Note that the coulomb and exchange operators depend on our initial guess.

The initial estimate of secular equation det 𝐹𝑘𝑙 − 𝜖𝑖𝑆𝑘𝑙 = 0 is

−0.813 − 𝜖𝑖 −0.892 − 0.8366𝜖𝑖

−0.892 − 0.8366𝜖𝑖 −0.070 − 𝜖𝑖= 0

𝜖1 = −0.854

𝜖2 = 2.885

Substitution of the 𝜖1 into the Roothan equation gives improved orbital

𝜓𝑖 = 0.836𝜙1 + 0.189𝜙2

After few cycles the converged roots from secular equation are 𝜖1 = −0.918 and 𝜖2 = 2.810 and the finalorbital is 𝜓𝑖 = 0.842𝜙1 + 0.183𝜙2. To obtain the ground state energy of the system one needs to add to the orbital energy (lower energy root 𝜖1) an additional core electron energy term and in the case of molecules, a nuclear-nuclear repulsion potential energy. So the SCF energy for helium becomes

𝐸HF = −0.918 − 1.944 + 0 = −2.862 hartrees

The limiting HF energy found with five basis functions is only 0.0000073 hartrees lower in energy. The comparison to variational and experimental energies shows an approximate error of 0.042 hartrees (110 kJ/mol), which arises due to lack of electron correlation.

• Input: Atom coordinates, the atomic number of the atoms, electron basis set

• Calculate Coulombic and exchange contributions

• Form Fock matrix using first guess orbitals

• Diagonalize Fock matrix and obtain improvedenergies and orbitals

• Repeat cycle until energies and orbitals remainunchanged

• Output: Energy, forces, electronic structure, etc.

Recall that the electron WF is characterized by spatial and spin variables

• If the number of electrons is even and orbitals are doublyoccupied, we have a closed shell (Fig. a)

• If the number of electrons is odd, we have an open-shellsystem (Fig. b)

• In general, if the numbers of electrons with spins up and down are different, we have an open-shell system

Open-shell systems can be dealt with by one of two HartreeFock methods:

• Restricted open-shell Hartree–Fock (ROHF)

• Unrestricted Hartree–Fock (UHF)

UHF theory is the most commonly used when the number of electrons of each spin are not equal. While RHF theory uses a single molecular orbital twice, UHF theory uses different molecular orbitals for the 𝛼 and 𝛽 electrons. The result is a pair of coupled Roothaan equations

𝐅𝛼𝐜𝛼 = 𝐒𝐜𝛼𝜖𝛼

𝐅𝛽𝐜𝛽 = 𝐒𝐜𝛽𝜖𝛽

The pair of equations are coupled because each orbital has to be optimised in the average field of all other electrons. This yields sets of molecular orbitals and orbital energies for both the 𝛼 and 𝛽 spin electrons.

UHF method has one drawback.

• A single Slater determinant of different orbitals for different spins is not a satisfactory eigenfunction of the total spin operator -𝐒2

• The electronic state can be contaminated by excited states. For example, the doublet state (one more 𝛼 spin electron than 𝛽 spin) may have too large total spin eigenvalue. If 𝐒2 = 0.8 or less

(exact value is 1

2

1

2+ 1 = 0.75), it is probably satisfactory. If it is 1.0 or so, it is certainly not

satisfactory and the calculation should be rejected and a different approach taken.

Given this drawback, why is the UHF used in preference to the ROHF?

• UHF is simpler to code

• it is easier to develop post-Hartree–Fock methods

• it is unique, unlike ROHF where different Fock operators can give the same final wave function.

• Large systematic error in HF electronicenergy

• The equilibrium structure and the shape of the potential energy surface near theequilibrium position is fairly well described

• The RHF method fails when atoms gain ioniccharacter

• The UHF can be used to calculate relativeenergy changes when molecule dissociates

Water dissociation

• Helium atom Hamiltonian

𝐻 = −∇1

2

2−

∇22

2−

2

𝑟1−

2

𝑟2+

1

𝑟12

• True wavefunction posesses a coulomb hole at 𝑟1 =𝑟2 and 𝛼 = 0.

• The HF wavefunction (expressed as a single Slaterdeterminant) has no special behaviour nearcoalescence, i.e. No electron correlation

• In reality, electrons tend to avoid each other.

• Unphysically large propabilities to finding twoelectrons near eachother results in overestimatedpotential energies in Hartree-Fock calculations

Lack of electron correlation in HF leads to an error of 1 eV (100 kJ/mol) per (valence)electron pair.

• In some cases the errors cancel themselves out

• Isodesmic reactions wherein the number and type of bonds on each side of the reaction remains unchanged can be calculated within few kcal/mol

Example: Reaction of ethanol with methane

• ethanol + methane→methanol + ethane

• The heat of formation of ethanol can be estimated now by simply calculating the reaction energy, with quantum mechanical methods and by using the computed reaction energy together with the known heats of formation.

• Using cheaply calculated HF/STO-3G energies for all four species, a reaction energy of +10.9 kJ/mol is predicted. Together with the know heats of formation, a value of -220 kJ/mol is predicted for ethanol. This has to be compared to the experimental value of -235.3 kJ/mol.

Hartree-Fock and experimental equilibrium bond lengths 𝑅𝑒 (in pm)

Molecule Bond HF Exp.

H2 𝑅HH 73.4 74.1

HF 𝑅FH 89.7 91.7

H2O 𝑅OH 94.0 95.7

O3 𝑅OO 119.2 127.2

CO2 𝑅CO 113.4 116

C2H4 𝑅CC 131.3 133.4

CH4 𝑅CH 108.2 108.6

• Hartree-Fock calculations systematically underestimate equilibrium bond lengths

• The HF results are satisfactory

Hartree-Fock and experimental electronic atomization energies (kJ/mol)

• Hartree-Fock calculations systematically underestimate atomization energies

• Hartree-Fock method fails to describe correctly the electronic structures of some molecules such as diatomic fluorine and ozone.

Molecule HF Exp.

F2 -155.3 163.4

H2 350.8 458.0

HF 405.7 593.2

H2O 652.3 975.3

O3 -238.2 616.2

CO2 1033.4 1632.5

C2H4 1793.9 2359.8

CH4 1374.1 1759.3

Hartree-Fock and experimental electronic reaction enthalpies (kJ/mol)

• Hartree-Fock method fails when reaction is far from isodesmic

• Some results are suprisingly good

Reaction HF Exp.

CO + H2 → CH2O 2.7 -21.8

H2O + F2 → HOF + HF -139.1 -129.4

N2+3H2→ 2NH3 -147.1 -165.4

C2H2+H2→ C2H4 -214.1 -203.9

CO2+4H2→ CH4 + 2H2O -242.0 -245.3

2CH2 → C2H4 -731.8 -845.7

O3+3H2→ 3H2O -1142.7 -935.5

Hartree-Fock summary

Electronic state described by single Slater determinant

• Electronic ground state

• Each electron move in mean field created by other electrons

• Electron correlation is largely neglegted which can lead to large deviations fromexperimental results.

Based on variational approach

• The best possible solution is at the Hartree-Fock limit: Energy as the basis set approaches completeness.

HF method provides the starting point for the methods that take the electron correlation into account (the post-HF methods). The model can be systematically improved by applying corrections.

• In HF the electronic wave function is approximated by a single Slaterdeterminant

• Not flexible enough to account for electron correlation

• therefore the Hartree-Fock limit is always above the exact energy

• Some electron correlation is already found in the electron exchange term

• Better description for the wavefunctionneeded for more accurate results

Electron correlation

Some common nomenclature found in literature:

Fermi correlation arises from the Pauli antisymmetry of the wave function and some of it is taken into account already at the single-determinant level.

For example the wavefunction for ground-state Helium is

𝜓 1,2 = 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 2 𝛽 2 − 𝜓1𝑠 2 𝛼 2 𝜓1𝑠 1 𝛽 1

where 1 and 2 in parentheses denote the coordinates of electrons 1 and 2, respectively. Clearly we see that if the coordinates of electrons have the same values, the wavefunctionvanishes

𝜓 1,2 = 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 1 𝛽 1 − 𝜓1𝑠 1 𝛼 1 𝜓1𝑠 1 𝛽 1 = 0

The wavefunction vanishes at the point where the two electron coincide. Around each electron there will be a hole in which there is less electron with a same spin: the Fermi hole. This ”exchange force” is comparatively localized.

Static correlation refers to situations in which multiple determinants are required to cover the coarse electronic structure. Electronic state is described by a combination of (qualitatively) different configurations which have comparative weights. Static correlations deals with only few, but very important determinants. Multi-configurational self-consistent field (MCSCF) is used to handle such correlations.

• Bond dissociation

• Excited states

• Near-degeneracy of electronic configurations (for example a singlet diradical CH2).

Dynamical correlation arises from Coulombic repulsion. To account for this, many determinants with small weights each are required. Dynamical correlation is needed to get the energetics of a system right, but not for the coarse electronic structure. Quite predictable, the major contribution is around 1 eV for each closed shell pair. Well accounted for by DFT functionals, perturbation theory, configuration interaction, and coupled cluster methods.

• Coulomb hole: The probability of finding two electrons at the same point in space is 0 as the repulsion becomes infinite. For the wavefunction approximated by Hartree-Fock method this requirement is not fulfilled.

There is no explicit separation between dynamical and static correlations.

Hartree-Fock wave function of He atom Exact wave function of He atom

the wave function of electron 1 while keeping the another fixed at x=0.5Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

nucleus nucleuselectron 2electron 2

Correlation hole

Difference between the exact and HF wave functions for the He atom

nucleuselectron 2

Configuration interaction (CI) has the following characteristics:

• A post-HartreeFock linear variational method.

• Solves the nonrelativistic Schrödinger equation within the BO–approximation for a multi-electron system.

• Describes the linear combination of Slater determinants used for the wave function.

o Orbital occupation (for instance, 1s22s21p1...) interaction means the mixing of different electronic configurations (states).

In contrast to the HartreeFock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state functions (CSFs) built from spin orbitals:

𝜓 =

𝑖

𝐶𝑖𝜓𝑖 = 𝐶0𝜓0 + 𝐶1𝜓1 + 𝐶2𝜓2 + ⋯

Configuration interaction method

Coefficients from the wavefunction expansion are determined by a variational optimization respect to the electronic energy

𝐇𝐂 = 𝐄𝐂𝐈𝐂

where H is the Hamiltonian matrix with matrix elements

𝐻𝑖𝑗 = 𝜓𝑖 𝐻 𝜓𝑗

The construction of the CI wavefunction may be carried out by diagonalization of the Hamiltonian matrix, but in reality iterative techniques are used to extract eigenvalues and eigenfunctions(Newton’s method).

The first term in the CI-expansion is normally the Hartree–Fock determinant

𝜓 = 𝐶0𝜓HF + 𝐶1𝜓1 + 𝐶2𝜓2 + ⋯

The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree–Fockdeterminant

• If only one spin orbital differs, we describe this as a single excitation determinant

• If two spin orbitals differ it is a double excitation determinant and so on

• The eigenvalues are the energies of the ground and some electronically excited states. By this it is possible to calculate energy differences (excitation energies) with CI-methods.

The expansion to the full set of Slater determinants (SD) or CSFs by distributing all electrons among all orbitals is called full CI (FCI) expansion. FCI exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set.

• In FCI, the number SDs increase very rapidly with the number of electrons and number of orbitals. For example, when distributing 10 electrons to 10 orbitals the number of SDs is 63504. This illustrates the intractability of the FCI for any but the smallest electronic systems.

• Practical solution: Truncation of the CI-expansion. Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. These methods, CID and CISD, are in many standard programs.

CI-expansion truncation is handled differently between static or dynamical correlation.

• In the treatment of static correlation in addition to the dominant configurations, near degenerate configurations are chosen (referred as reference configurations).

• Dynamical correlation is subsequently treated by generating excitations from reference space.

Excitation energies of truncated CI-methods are generally too high because the excited states are not that well correlated as the ground state is.

• The Davidson correction can be used to estimate a correction to the CISD energy to account for higher excitations. It allows one to estimate the value of the full configuration interaction energy from a limited configuration interaction expansion result, although more precisely it estimates the energy of configuration interaction up to quadruple excitations (CISDTQ) from the energy of configuration interaction up to double excitations (CISD). It uses the formula:

δ𝐸𝑄 = 1 − 𝐶02 𝐸CISD − 𝐸HF

where 𝐶0 is the coefficient of the Hartree–Fock wavefunction in the CISD-expansion

CI-methods are not size-consistent and size-extensive

• Size-inconsistency means that the energy of two infinitely separated particles is not double the energy of the single particle. This property is of particular importance to obtain correctly behaving dissociation curves.

• Size-extensivity, on the other hand, refers to the correct (linear) scaling of a method with the number of electrons.

• The Davidson correction can be used.

• Quadratic configuration interaction (QCI) is an extension CI that corrects for size-consistency errors in the all singles and double excitation CI methods (CISD). This method is linked to coupled cluster (CC) theory.

o Accounts for important four-electron correlation effects by including quadruple excitations

Example: CI calculation for Helium atom

Lets us begin with two-configuration wavefunction expressed as a linear combination of hydrogenicwavefunctions having the form

𝜓1,2 = 𝑐1𝜓1 + 𝑐2𝜓2

where 𝜓1 arises from the configuration 1𝑠2 and 𝜓2 arises from the configuration 1𝑠2𝑠. Specifically, the two wavefunctions are

𝜓1 =1

2

1𝑠 1 𝛼 1 1𝑠 1 𝛽 1

1𝑠 2 𝛼 2 1𝑠 2 𝛽 2

𝜓2 =1

2

1𝑠 1 𝛼 1 1𝑠 1 𝛽 1

2𝑠 2 𝛼 2 2𝑠 2 𝛽 2+

1

2

2𝑠 1 𝛼 1 2𝑠 1 𝛽 1

1𝑠 2 𝛼 2 1𝑠 2 𝛽 2

Since both 𝜓1 and 𝜓2 describe singlet states, there will be no vanishing matrix elements of 𝐻.

If we represent these matrix elements by 𝐻𝑖𝑗 = 𝜓𝑖 𝐻 𝜓𝑗 (𝑖, 𝑗 = 1 or 2), the secular

determinant to be solved is

𝐻11 − 𝐸 𝐻12

𝐻12 𝐻22 − 𝐸= 0

The diagonal matrix elements 𝐻11 and 𝐻22 are just the energies of single configurational calculations for the ground and exited states. Keeping in mind that the spin portions integrate out separately to unity 𝛼 𝛼 = 𝛽 𝛽 = 1 , 𝛼 𝛽 = 0 we obtain for the 𝐻11

𝐻11 = 1𝑠 1 1𝑠 2 𝐻 1𝑠 1 1𝑠 2 = 2𝜖1 + 𝐽11

where we represent the 1𝑠 orbital by subscript 1. The energy 𝜖 has the same form as the hydrogen atom energy

𝜖𝑛 = −𝑍2

2𝑛2 = −2

𝑛2

Similarly we can obtain the expressions for the matrix elements 𝐻22 and 𝐻12 but this derivationis omitted here for the sake of simplicity and only the final results are shown. After the appropriate integrations, the matrix elements become

𝐻11 = −2.75

𝐻22 = −2.037

𝐻12 = 0.253

The roots of the quadratic formula (produced by secular determinant) are 𝐸1 = −2.831 and 𝐸2 = −1.956. The lower root represents the ground state whose experimental energy is −2.903. Note that this improves the single configurational result −2.75. The higher root represents the lowest excited state (experimental energy = −2.15).

Compared to CISD-method, the simpler and less computationally expensive MP2-method gives superior results when size of the system increases (MP2 is size extensive).

• For water monomer, MP2 recovers 94% of correlation energy which remains similar with increasing system (cc-pVTZ basis).

• For stretched water monomer (bond length doubled) CISD recovers only 80.2% of the correlation energy.

o for a large variety of systems it recovers 80-90% of the available correlation energy

o With the Davidson correction added, the error is reduced to 3%.

• When the number of monomers increases, the degradation in the performance is even more severe for the equilibrium geometry.

o For eight monomers, the CISD wavefunction recovers only half of the correlation energy and the Davidson correction remain more or less the same.

Dissociation of a water molecule Thick line: Full CIOne the right: Difference betweentruncated CI and FCI

Helgaker, Jorgensen, Olsen:Molecular Electronic-Structure Theory(Wiley 2002)

MRCISD wave functions in description of dissociation of a water molecule

Helgaker, Jorgensen, Olsen:Molecular Electronic-Structure Theory(Wiley 2002)

Difference between MRCISD and FCI

In the variational method the starting point is to try to guess the wave function of the system.Perturbation theory is not variational. It is based on the idea that exact solutions for a system thatresembles the true system are known. The solution for the real system is found by observing howthe Hamiltonian for the two systems differ.

Perturbation theory

In the picture (a) are shown the energy levels of harmonic oscillator. Its eigenfunctions and energy levels are known. In the picture (b) the system is perturbed by known potential energy term V’ which is the difference between potential energy curves of systems (a) and (b). The eigenfunctions and energy levels of system (b) are not known. The perturbation theory is devised to found approximate solutions for the properties of the perturbed system.

The Schrödinger equation for the perturbed state n is

𝐻𝜓𝑛 = 𝐻 0 + 𝜆𝑉 𝜓𝑛 = 𝐸𝑛𝜓𝑛

where 𝜆 is an arbitrary real parameter, 𝑉 is a perturbation to the unperturbed Hamiltonian 𝐻 0 , and subscript n = 1, 2, 3, ... denotes different discrete states.

• The expressions produced by perturbation theory are not exact

• Accurate results can be obtained as long as the expansion parameter 𝜆 is very small.

We expand 𝜓𝑛 and 𝐸𝑛 in Taylor series in powers of 𝜆. The eigenvalue equation becomes

𝐻 0 + 𝜆𝑉

𝑖=0

𝜆𝑖𝜓𝑛𝑖

=

𝑖=0

𝜆𝑖𝐸𝑛𝑖

𝑖=0

𝜆𝑖𝜓𝑛𝑖

Writing only the first terms we obtain

𝐻 0 + 𝜆𝑉 𝜓𝑛0

+ 𝜆𝜓𝑛1

= 𝐸𝑛0

+ 𝜆𝐸𝑛1

𝜓𝑛0

+ 𝜆𝜓𝑛1

The zeroth-order equation is simply the Schrödinger equation for the unperturbed system

𝐻 0 𝜓𝑛0

= 𝐸𝑛0

𝜓𝑛0

The first-order terms are those which are multiplied by 𝜆

𝐻 0 𝜓𝑛1

+ 𝑉𝜓𝑛0

= 𝐸𝑛0

𝜓𝑛1

+ 𝐸𝑛1

𝜓𝑛0

When this is multiplied through by 𝜓𝑛0 ∗

from left and integrated, the first term on the left-hand

side cancels with the first term on the right-hand side (The 𝐻 0 is hermitian). This leads to the first-order energy shift:

𝐸𝑛1

= 𝜓𝑛0

𝑉 𝜓𝑛0

This is simply the expectation value of the perturbation Hamiltonian while the system is in the

unperturbed state. The energy of the 𝑛th state up to the first order is thus 𝐸𝑛0

+ 𝐸𝑛1

.

Interpretation of the first order correction to energy:

𝐸𝑛1

= 𝜓𝑛0

𝑉 𝜓𝑛0

• The perturbation is applied, but we keep the system in the quantum state 𝜓𝑛0

, which is a valid quantum state though no longer an energy eigenstate.

• The perturbation causes the average energy of this state to increase by 𝜓𝑛0

𝑉 𝜓𝑛0

Further shifts are given by the second and higher order corrections to the energy.

To obtain the first-order correction to the energy eigenstate, we recall the expression derived earlier

𝐻 0 𝜓𝑛1

+ 𝑉𝜓𝑛0

= 𝐸𝑛0

𝜓𝑛1

+ 𝐸𝑛1

𝜓𝑛0

and multiply it by 𝜓𝑚0 ∗

, 𝑚 ≠ 𝑛 from left and integrate. We obtain

𝐸𝑚0

− 𝐸𝑛0

𝜓𝑚0

𝜓𝑛1

= − 𝜓𝑚0

𝑉 𝜓𝑛0

Multiplying from left with 𝑚≠𝑛 𝜓𝑚0

and using the resolution of identity, we obtain

𝜓𝑛1

=

𝑚≠𝑛

𝜓𝑚0

𝑉 𝜓𝑛0

𝐸𝑛0

− 𝐸𝑚0

𝜓𝑚0

The first-order change in the 𝑛-th energy eigenfunction has a contribution from each of the energy eigenstates 𝑚 ≠ 𝑛.

The second order correction to the energy is

𝐸𝑛2

=

𝑚≠𝑛

𝜓𝑚0

𝑉 𝜓𝑛0

2

𝐸𝑛0

− 𝐸𝑚0

Conclusion:

• Each term is proportional to the matrix element 𝜓𝑚0

𝑉 𝜓𝑛0

, that is a measure of how much the

perturbation mixes state 𝜓𝑛0

with state 𝜓𝑚0

• Correction is inversely proportional to the energy difference between states 𝜓𝑛0

and 𝜓𝑚0

, which means that the perturbation deforms the state to a greater extent if there are more states at nearby energies.

• Expression is singular if any of these states have the same energy as state 𝜓𝑛0

, which is why we assumed that there is no degeneracy

• Higher-order deviations can be found by a similar procedure

Example: PT calculation for Helium atom

Helium atom Hamiltonian

𝐻 = −∇1

2

2−

∇22

2−

2

𝑟1−

2

𝑟2+

1

𝑟12= ℎ1 + ℎ2 +

1

𝑟12

It is convinient to choose the unperturbed system a a two-electron atom in which the electrons do notinteract. The zeroth_order Hamiltonian then is

𝐻 0 = ℎ1 + ℎ2

The zeroth-order eigenfunctions have the hydrogen atom form

𝜓 0 = 1𝑠 1 1𝑠 2 =1

𝜋

𝑍

𝑎0

3 2

𝑒− 𝑍 𝑟1+𝑟2 𝑎0

And the zeroth-order energy is simply

𝐸 0 = −𝑍2 = −4.0

The first-order perturbation-energy correction is

𝐸 1 = 1𝑠 1 1𝑠 21

𝑟121𝑠 1 1𝑠 2 =

5

8𝑍 =

5

4

Thus, the total energy of the helium atom (to the first order) is

𝐸 = 𝐸 0 + 𝐸 1 = −4 +5

4= −2.75

This result is equal to the single configurational energy in previous CI calculation. Thus, electroncorrelation does not seem to contribute to the first order energy correction.

The total energy up to the second order is

𝐸 = 𝐸 0 + 𝐸 1 + 𝐸 2 = −2.75 − 0.157666405 = −2.90767

which is lower than true ground state energy −2.90372.

Introduction:

The Møller–Plesset perturbation theory (MP) was published as early as 1934 by Christian Møllerand Milton S. Plesset.

• The starting point is eigenfunction of the Fock-operator.

• It improves on the Hartree–Fock method by adding electron correlation effects.

• MP theory is not variational. Calculated energy may be lower than true ground state energy.

MP methods (MP2, MP3, MP4, ...) are implemented in many computational chemistry codes. Higher level MP calculations, generally only MP5, are possible in some codes. However, they are rarely used because of their costs.

Møller–Plesset perturbation theory

The MP-energy corrections are obtained with the perturbation 𝑉, which is defined as a difference between the true nonrelativistic Hamiltonian 𝐻 and the sum of one-electron Fock operators 𝐹

𝑉 = 𝐻 − 𝐹

The Slater determinant 𝜓 is the eigenfunction of the Fock-operator 𝐹

𝐹𝜓 =

𝑖

𝜖𝑖 𝜓

where 𝜖𝑖 is the orbital energy belonging to the doubly occupied space orbital.

The sum of MP zeroth-order energy and first order energy correction is

𝐸𝑀𝑃(0)

+ 𝐸𝑀𝑃(1)

= 𝜓 𝐹 𝜓 + 𝜓 𝑉 𝜓 = 𝜓 𝐻 𝜓

But 𝜓 𝐻 𝜓 is the variational integral for the Hartree-Fock wave function 𝜓 and it therefore equals the Hartree-Fock energy 𝐸𝐻𝐹.

𝐸𝑀𝑃(0)

+ 𝐸𝑀𝑃(1)

= 𝐸𝐻𝐹

In order to obtain the MP2 formula for a closed-shell molecule, the second-order correction formula is written on basis of doubly-excited Slater determinants (singly-excited Slater determinants vanish).

𝐸𝑀𝑃(2)

= 𝑖>𝑗𝑎>𝑏

𝜙𝑎 1 𝜙𝑏 2 𝑟12−1 𝜙𝑖 1 𝜙𝑗 2 − 𝜙𝑎 1 𝜙𝑏 2 𝑟12

−1 𝜙𝑗 1 𝜙𝑖 22

𝜖𝑖 + 𝜖𝑗 − 𝜖𝑎 − 𝜖𝑏

where 𝜙𝑖 and 𝜙𝑗 are occupied orbitals and 𝜙𝑎 and 𝜙𝑏 are virtual (unoccupied) orbitals. The quantities 𝜖𝑖, 𝜖𝑗, 𝜖𝑎, and 𝜖𝑏

are the corresponding orbital energies. Up to the second-order, the total electronic energy is given by the Hartree–Fockenergy plus second-order MP correction:

𝐸 = 𝐸𝐻𝐹 + 𝐸𝑀𝑃(2)

Calculated and experimental atomization energies (kJ/mol)

• Accuracy of the MP2 is satisfactory despite its relatively low computational cost

• MP2 usually overestimates bond energies

Molecule HF MP2 Exp.

F2 -155.3 185.4 163.4

H2 350.8 440.7 458.0

HF 405.7 613.8 593.2

H2O 652.3 996.1 975.3

O3 -238.2 726.6 616.2

CO2 1033.4 1745.2 1632.5

C2H4 1793.9 2379.3 2359.8

CH4 1374.1 1753.1 1759.3

Calculated and experimental reaction enthalpies (kJ/mol)

• The accuracy of MP2 is much improved compared to HF

• It is problematic to improve MP calculations systematically

Reaction HF MP2 Exp.

CO + H2 → CH2O 2.7 -25.0 -21.8

H2O + F2 → HOF + HF -139.1 -127.2 -129.4

N2+3H2→ 2NH3 -147.1 -164.4 -165.4

C2H2+H2→ C2H4 -214.1 -196.1 -203.9

CO2+4H2→ CH4 + 2H2O -242.0 -237.3 -245.3

2CH2 → C2H4 -731.8 -897.9 -845.7

O3+3H2→ 3H2O -1142.7 -939.7 -935.5

Dissociation of a water molecule Thick line: FCIFull line: RHF reference stateDashed line: UHF reference stateOne the right: Difference between MPPT and FCI

Helgaker, Jorgensen, Olsen:Molecular Electronic-Structure Theory(Wiley 2002)

Concluding remarks

• Systematic studies of MP perturbation theory have shown that it is not necessarily a convergent theory at high orders. The convergence properties can be slow, rapid, oscillatory, regular, highly erratic or simply non-existent, depending on the precise chemical system or basis set.

• Various important molecular properties calculated at MP3 and MP4 level are in no way better than their MP2 counterparts, even for small molecules.

• For open shell molecules, MPn-theory can directly be applied only to unrestricted Hartree–Fockreference functions. However, the resulting energies often suffer from severe spin contamination, leading to very wrong results. A much better alternative is to use one of the MP2-like methods based on restricted open-shell Hartree–Fock references.

First observations:

• Coupled cluster (CC) method, especially The CCSD(T), has become the ”gold-standard of quantum chemistry”. CC theory was poised to describe essentially all the quantities of interest in chemistry, and has now been shown numerically to offer the most predictive, widely applicable results in the field.

• The computational cost is very high. So, in practice, it is limited to relatively small systems.

Some facts:

• Coupled cluster (CC) is a numerical technique used for describing many-body systems.

• It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation.

Coupled cluster method

• The CC method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics.

• In 1966 Jiri Cek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules.

• Kümmel comments:

I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Jiri’s work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then.

Some history:

The wavefunction of the coupled-cluster theory is written in terms of exponential functions:

𝜓 = 𝑒 𝑇 𝜙0

Where is a Slater determinant usually constructed from Hartree–Fock molecular orbitals. The operator 𝑇 is an excitation operator which, when acting on 𝜙0 , produces a linear combination of excited Slater determinants.

The exponential approach guarantees the size extensivity of the solution. For two subsystems A and B and corresponding excitation operators 𝑇A and 𝑇B, the exponential function admits for the simple

factorization 𝑒 𝑇A+ 𝑇B = 𝑒 𝑇A𝑒 𝑇B. Therefore, aside from other advantages, the CC method maintains the property of size consistency.

The cluster operator is written in the form

𝑇 = 𝑇1 + 𝑇2 + 𝑇3 + ⋯

where 𝑇1 is the operator of all single excitations, 𝑇2 is the operator of all double excitations and so forth.

The exponential operator 𝑒 𝑇 may be expanded into Taylor series:

𝑒 𝑇 = 1 + 𝑇 + 𝑇2

2!+ ⋯ = 1 + 𝑇1 + 𝑇2 +

𝑇12

2+ 𝑇1

𝑇2 + 𝑇2

2

2+ ⋯

In practice the expansion of 𝑇 into individual excitation operators is terminated at the second or slightly higher level of excitation. Slater determinants excited more than 𝑛 times contribute to the wave function because of the non-linear nature of the exponential function. Therefore, coupled cluster terminated at 𝑇𝑛usually recovers more correlation energy than CI with maximum 𝑛excitations.

A drawback of the method is that it is not variational

𝐸 𝜙 =𝜙 𝑒− 𝑇𝐻𝑒 𝑇 𝜙

𝜙 𝜙

which for truncated cluster expansion becomes

𝐸 Φ =𝜒 𝐻 𝜊

𝜙 𝜙

where 𝜒 and 𝜊 are different functions

The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of 𝑇. The abbreviations for coupled-cluster methods usually begin with the letters CC (for coupled cluster) followed by

• S - for single excitations (shortened to singles in coupled-cluster terminology)

• D - for double excitations (doubles)

• T - for triple excitations (triples)

• Q - for quadruple excitations (quadruples)

Thus, the operator in CCSDT has the form:

𝑇 = 𝑇1 + 𝑇2 + 𝑇3

Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, a CCSD(T) approach simply means:

• It includes singles and doubles fully

• Triples are calculated with perturbation theory.

Calculated and experimental atomization energies (kJ/mol)

• CCSD calculations produce qualitatively correct result

• Eventhought CCSD is expensive method, it is unfortunately not very accurate

Molecule HF CCSD MP2 Exp.

F2 -155.3 128.0 185.4 163.4

H2 350.8 458.1 440.7 458.0

HF 405.7 583.9 613.8 593.2

H2O 652.3 960.2 996.1 975.3

O3 -238.2 496.1 726.6 616.2

CO2 1033.4 1573.6 1745.2 1632.5

C2H4 1793.9 2328.9 2379.3 2359.8

CH4 1374.1 1747.0 1753.1 1759.3

Calculated and experimental reaction enthalpies (kJ/mol)

• CCSD recovers majority of the electron correlation energy

• CCSD calculations do not achieve chemical accuracy (4 kJ/mol)

Reaction HF CCSD MP2 Exp.

CO + H2 → CH2O 2.7 -23.4 -25.0 -21.8

H2O + F2 → HOF + HF -139.1 -123.3 -127.2 -129.4

N2+3H2→ 2NH3 -147.1 -173.1 -164.4 -165.4

C2H2+H2→ C2H4 -214.1 -209.7 -196.1 -203.9

CO2+4H2

→ CH4 + 2H2O-242.0 -261.3 -237.3 -245.3

2CH2 → C2H4 -731.8 -830.1 -897.9 -845.7

O3+3H2→ 3H2O -1142.7 -1010.1 -939.7 -935.5

Deviation of CI and CC energies from non-relativistic exact results(within B-O approximation) for H2O (mHartree = 2.6255 kJ/mol)

• CI converges (too) slowly to exact energy

• CC has superior performance but show fluctuations

Method 𝒓𝐞 1.5 𝒓𝐞 2 𝒓𝐞

Hartree Fock 216.1 270.9 370.0

CID 13.7 34.5 84.8

CISD 12.9 30.4 75.6

CISDT 10.6 23.5 60.3

CISDTQ 0.40 1.55 6.29

CCD 5.01 15.9 40.2

CCSD 4.12 10.2 21.4

CCSDT 0.53 1.78 -2.47

CCSDTQ 0.02 0.14 -0.02

We have learned that:

• Ab initio methods based on coupled cluster (CC) approach, are currently the most precise tool to calculate electron correlation effects.

• Some of the most accurate calculations for small to medium sized molecules use this method.

• The computational cost is very high. So, in practice, it is limited to relatively small systems.

• A drawback of the method is that it is not variational

• Unfortunately, for high precision work, the CCSD model is usually not accurate enough and CCSDT model is too expensive.

The solution:

• We can combine the CC and perturbation theory. Let the PT take care of the computationally expensive high excitation terms. The acronym for the most popular hybrid method is CCSD(T) where T in the brackets means perturbative triple excitations.

• The CCSD(T) method has become the ”gold-standard of quantum chemistry” as it very reliably reaches the so-called chemical accuracy (energies within about 1-2 kcal/mol (1 kcal = 4.18 kJ) of experimental values) when computing molecular properties for wide range of chemical species.

Example: Water molecule CCSD(T) calculations

• For OH bond distances less than 3.5Å, the CCSD(T) works well, giving about 90% of the full CCSDT triples correction.

• The model breaks down at larger OH bond distances. The unrestricted CCSD(T) (based on UHF reference) does not provide good description of the dissociation process.

Calculated and experimental electronic atomization energies (kJ/mol)

• CCSD(T) calculations produce accurate results

• Only for most problematic systems (such as ozone) higher order corrections aredesirable

Molecule HF CCSD CCSD(T) Exp.

F2 -155.3 128.0 161.1 163.4

H2 350.8 458.1 458.1 458.0

HF 405.7 583.9 593.3 593.2

H2O 652.3 960.2 975.5 975.3

O3 -238.2 496.1 605.5 616.2

CO2 1033.4 1573.6 1633.2 1632.5

C2H4 1793.9 2328.9 2360.8 2359.8

CH4 1374.1 1747.0 1759.4 1759.3

Calculated and experimental electronic reaction enthalpies (kJ/mol)

• CCSD(T) is generally an improvement over CCSD

• CCSD(T) models chemical reactions mostly within chemical accuracy (4 kJ/mol)

Reaction CCSD CCSD(T) Exp.

CO + H2 → CH2O -23.4 -23.0 -21.8

H2O + F2 → HOF + HF -123.3 -119.5 -129.4

N2+3H2→ 2NH3 -173.1 -165.5 -165.4

C2H2+H2→ C2H4 -209.7 -205.6 -203.9

CO2+4H2→ CH4 + 2H2O -261.3 -244.7 -245.3

2CH2 → C2H4 -830.1 -844.9 -845.7

O3+3H2→ 3H2O -1010.1 -946.6 -935.5

Error in the reaction enthalpies (kJ/mol) for 14 reactions involving small main-group element molecules

Dissociation of a water molecule Full line: RHF reference state, dashed line: UHF reference state

Helgaker, Jorgensen, Olsen:Molecular Electronic-Structure Theory(Wiley 2002)

Comparison of models by the deviation from experimentalmolecular geometries of 29 small main-group element species

cc-pVDZ cc-pVTZ

Difference to the FCI energy of various CC and MP levels of theory. Water molecule in equilibrium and stretched geometries.

Relationship between the calculated bond distances for the standard models (in pm)

Performance vs. accuracy of different ab initio methods

Level of theory Equilibriumgeometry

Equilibrium and stretched geometry

Scaling

MP2 10.4 17.4 𝑁5

MP3 5.0 14.4 𝑁6

CISD 5.8 13.8 𝑁6

CCD 2.4 8.0 𝑁6

CCSD 1.9 4.5 𝑁6

QCISD 1.7 4.0 𝑁6

MP4 1.3 3.7 𝑁7

CCSD(T) 0.3 0.6 𝑁7

QCISD(T) 0.3 0.5 𝑁7

CCSDT 0.2 0.5 𝑁8

CCSDTQ 0.01 0.02 𝑁10

Average errors in correlation energies compared to full CI applied to HB, HF, and H2O at bothequilibrium and bond-stretched geometries. Scaling describes a computational cost respect to sizeof the system (number of electrons and basis functions). Hartree-Focks formally scales as 𝑁4.

A basis set is a set of functions used to create the molecular orbitals, which are expanded as a linear combination with coefficients to be determined.

• Usually these functions are centered on atoms, but functions centered in bonds or lone pairs have been used.

• Additionally, basis sets composed of sets of plane waves are often used, especially in calculations involving systems with periodic boundary conditions (continuous systems, surfaces).

Quantum chemical calculations are typically performed within a finite set of basis functions.

• These basis functions are usually not the exact atomic orbitals, like the hydrogen atom eigenfunctions.

• If the finite basis is expanded towards an infinite complete set of functions, calculations using such a basis set are said to approach the basis set limit.

Basis sets

In the early days of quantum chemistry so-called Slater type orbitals (STOs) were used as basis functions due to their similarity with the eigenfunctions of the hydrogen atom. Their general definition is

𝜓𝑛𝑙𝑚 𝑟, 𝜃, 𝜙 = 𝑁𝑟𝑛−1𝑒− 𝜁𝑟 𝑎0𝑌𝑙𝑚 𝜃, 𝜙

where 𝑛 = 1,2, … is related to hydrogen atom principal quantum number, and 𝑙 and 𝑚 are related to hydrogen atom angular momentum and magnetic quantum numbers, respectively. 𝑁 is a normalization factor, 𝜁 is the effective nuclear charge, and 𝑌𝑙

𝑚 𝜃, 𝜙 being the spherical harmonics.

• STOs have an advantage in that they have direct physical interpretation and thus are naturally good basis for molecular orbitals.

• From a computational point of view the STOs have the severe shortcoming that most of the required integrals needed in the course of the SCF procedure must be calculated numerically which drastically decreases the speed of a computation.

• Still, today there exist some modern and efficient computational chemistry program packages that use STOs (ADF).

STOs can be approximated as linear combinations of Gaussian type orbitals, which are defined as

𝑔𝑖𝑗𝑘 𝐫 = 𝑁 𝑥 − 𝑅𝑥𝑖 𝑦 − 𝑅𝑦

𝑗𝑧 − 𝑅𝑧

𝑘𝑒−𝛼 𝐫−𝐑 2

𝑁 is a normalization factor, 𝐑 is the atomic center, and 𝛼 is an orbital exponent of the Gaussian function, respectively. GTOs are not really orbitals, they are simpler functions (Gaussian primitives).

GTOs are usually obtained from quantum calculations on atoms (i.e. Hartree-Fock or Hartree-Fockplus some correlated calculations, e.g. CI).

• Typically, the exponents 𝛼 are varied until the lowest total energy of the atom is achieved.

• For molecular calculations, certain linear combinations of GTOs will be used as basis functions.

• Such a basis function (contraction) will have its coefficients and exponents fixed. For example:

𝜙1 = 𝑎𝑔1 + 𝑏𝑔2 + 𝑐𝑔3

Where coefficients 𝑎, 𝑏, and 𝑐 and the exponents 𝛼 in functions 𝑔 are fixed (i.e. are not variables).

The main difference to the STOs is that the variable 𝑟 in the exponential function 𝑒−𝛼 𝐫−𝐑 2is

squared. Generally the inaccuracy at the center or the qualitatively different behaviour at long distances from the center have a marked influence on the results.

The radial parts of the orbitals plotted in the figures

To understand why integrals over GTOs can be carried out when analogous STO-based integrals are much more difficult, one must consider orbital products 𝜓a, 𝜓b, 𝜓c, and 𝜓d where a, b, c, and d refer to different atomic centers. These products give rise to multi-center integrals, which often arise in polyatomic-molecule calculations, and which can not be efficiently performed when STOs are employed. For orbitals in the GTO form, can be rewritten as

𝑒−𝛼a 𝐫−𝐑a2𝑒−𝛼c 𝐫−𝐑c

2= 𝑒− 𝛼a+𝛼c 𝐫−𝐑′ 2

𝑒−𝛼′ 𝐑a−𝐑c2

where

𝐑′ =𝛼a𝐑a + 𝛼c𝐑c

𝛼a + 𝛼c

and

𝛼′ =𝛼a𝛼c

𝛼a + 𝛼c

Thus, the product of two GTOs on different centers is equal to a single other GTO at center R′ between the original centers. As a result, even a four-center integral over GTOs can be written as two-center two-electron integral. A similar reduction does not arise for STOs.

In GTOs 𝑁 𝑥 − 𝑅𝑥𝑖 𝑦 − 𝑅𝑦

𝑗𝑧 − 𝑅𝑧

𝑘𝑒−𝛼 𝐫−𝐑 2the sum of the exponents of the cartesian

coordinates, 𝐿 = 𝑖 + 𝑗 + 𝑘, is used to mark functions as 𝑠-type (𝐿 = 0), 𝑝-type (𝐿 = 1), 𝑑-type (𝐿 = 2), and so on

• Unfortunately GTOs are not eigenfunctions of the squared angular momentum operator 𝐿2.

• However, combinations of GTOs are able to approximate correct nodal properties of atomic orbitals by taking them with different signs. For example combining three 𝑑-type cartesian GTOs yields a cartesian GTO of 𝑠-type:

𝑔200 + 𝑔020 + 𝑔002 ∝ 𝑔000

• Today, there are hundreds of basis sets composed of GTOs. The smallest of these are called minimal basis sets, and they are typically composed of the minimum number of basis functions required to represent all of the electrons on each atom. The largest of these can contain literally dozens to hundreds of basis functions on each atom.

A minimum basis set is one in which a single basis function is used for each orbital in a Hartree-Fockcalculation on the free atom.

• The most common minimal basis set is STO-𝑛G, where 𝑛 is an integer. This n value represents the number GTOs used to approximate STO for both core and valence orbitals.

• Minimal basis sets typically give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts.

• Commonly used minimal basis sets of this type are: STO-3G, STO-4G, STO-6G

The minimal basis sets are not flexible enough for accurate representation of orbitals

Solution: Use multiple functions to represent each orbital

For example, the double-zeta basis set allows us to treat each orbital separately when we conduct the Hartree-Fock calculation.

𝜓2𝑠 𝐫 = 𝑐1𝜓2𝑠 𝐫; 𝜁1 + 𝑐2𝜓2𝑠 𝐫; 𝜁2

where 2𝑠 atomic orbital is expressed as the sum of two STOs. The 𝜁-coefficients account for how large the orbitals are. The constants 𝑐1 and 𝑐2 determine how much each STO will count towards the finalorbital.

The triple and quadruple-zeta basis sets work the same way, except use three and four Slater equations (linear combination of GTOs) instead of two. The typical trade-off applies here as well, better accuracy...more time/work.

Pople’s split-valence basis sets n-ijG or n-ijkG.

• n - number of GTOs for the inner shell orbitals; ij or ijk– number of GTOs for basis functions in the valence shell. The ij notations describes sets of valence doublezeta quality and ijk sets of valence triple zeta quality.

• The 𝑠-type and 𝑝-type functions belonging to the sameelectron shell are folded into a 𝑠𝑝-shell. In this case, number of 𝑠-type and 𝑝-type GTOs is the same, and they have identical exponents. However, thecoefficients for 𝑠-type and 𝑝-type basis functions aredifferent.

Example: Four 𝑠-type GTOs used to represent 1𝑠 orbital of hydrogen as:

𝜓1𝑠 = 0.50907𝑁1𝑒−𝛼1𝑟2

+ 0.47449𝑁2𝑒−𝛼2𝑟2

+ 0.13424𝑁3𝑒−𝛼3𝑟2

+ 0.01906𝑁4𝑒−𝛼

4𝑟2

where 𝑁𝑖 is a normalization constant for a given GTO and 𝛼𝑖 are the exponents. These GTOs may be grouped in 2 basis functions. The first basis function contains only 1 GTO:

𝜙1 = 𝑁1𝑒−𝛼1𝑟2

3 GTOs are present in the second basis function:

𝜙2 = 𝑁 0.47449𝑒−𝛼2𝑟2

+ 0.13424𝑒−𝛼3𝑟2

+ 0.01906𝑒−𝛼4𝑟2

where 𝑁 is a normalization constant for the whole basis function. In this case, 4 GTOs were contracted to 2 basis functions. It is frequently denoted as 4𝑠 → 2𝑠 contraction. The coefficients in function are then fixed insubsequent molecular calculations.

Example: Silicon 6-31G basis set

• The corresponding exponents for 𝑠-type and 𝑝-type basis functions are equal but coefficients in 𝑠-type and 𝑝-type basis functions are different.

• GTOs are normalized here since coefficients for basis functions consisting of one GTO (last row) are exactly 1.

• The basis set above represents the followingcontraction 16𝑠, 10𝑝 → 4𝑠, 3𝑝

Example: 3-21G basis set of carbon Surface and contour plot of p–type basis function including two Gaussians.

𝜙2𝑝,1: =2 ∗ 𝛼sp,1

𝜋

34

∗ e−𝛼sp,1 𝑥2+𝑦2+𝑧2

𝜙2𝑝𝑦: = 𝑦 ∗ 𝑐2𝑝,1 ∗ 𝜙2𝑝,1 + 𝑐2𝑝,2 ∗ 𝜙2𝑝,2

𝛼sp,1: = 3.664980

𝛼sp,2 ≔ 0.770545

𝑐2𝑝,1: = 0.236460

𝑐2𝑝,2: = 0.860619

Polarized basis sets

• Polarization functions denoted in Pople’s sets by an asterisk

• Two asterisks, indicate that polarization functions are also added to light atoms (hydrogen and helium).

• Polarization functions have one additional node.

• For example, the only basis function located on a hydrogen atom in a minimal basis set would be a function approximating the 1𝑠 atomic orbital. When polarization is added to this basis set, a 𝑝-type function is also added to the basis set.

6-31G**

Polarization functions add flexibility within the basis set, effectively allowing molecular orbitals to be more asymmetric about the nucleus.

• This is an important for accurate description of bonding between atoms, because the precence of the other atom distorts the environment of the electrons and removes its spherical symmetry.

• Similarly, 𝑑-type functions can be added to a basis set with valence 𝑝-type orbitals, and so on.

• High angular momentum polarization functions (𝑑, 𝑓, …) are important for heavy atoms

Some observations concerning polarization functions:

• The exponents for polarization functions cannot be derived from Hartree-Fock calculations for the atom, since they are not populated.

• In practice, these exponents are estimated ”using well established rules of thumb” or by using a test set of molecules.

• The polarization functions are important for reproducing chemical bonding.

• They should be included in all calculations where electron correlation is important.

• Adding them is costly. Augmenting basis set with 𝑑-type polarization functions adds 5 basis function on each atom while adding 𝑓-type functions adds 7.

The basis sets are also frequently augmented with the so-called diffuse functions.

• These Gaussian functions have very small exponents and decay slowly with distance from thenucleus.

• Diffuse gaussians are usually of 𝑠-type and 𝑝-type.

• Diffuse functions are necessary for correct description of anions and weak bonds (e.g. hydrogen bonds) and are frequently used for calculations of properties (e.g. dipole moments, polarizabilities, etc.).

For the Pople’s basis sets the following notaton is used:

• n-ij+G, or n-ijk+G when 1 diffuse 𝑠-type and 𝑝-type gaussian with the same exponents are added to a standard basis set on heavy atoms.

• The n-ij++G, or n-ijk++G are obtained by adding 1 diffuse 𝑠-type and 𝑝-type gaussian on heavy atoms and 1 diffuse 𝑠-type gaussian on hydrogens.

The area which is modelled by diffuse functions.

Diffuse functions are very shallow Gaussian basis functions, which more accurately represent the ”tail” portion of the atomic orbitals, which are distant from the atomic nuclei.

Diffuse functions

Correlation consistent basis sets are widely used basis sets are those developed by Dunning and co. These basis sets have become the current state of the art for correlated calculations

• Designed to converge systematically to the complete basis set (CBS) limit using extrapolationtechniques

• For first- and second-row atoms, the basis sets are cc-pVnZ where n=D,T,Q,5,6,... (D=double-zeta, T=triple-zeta, etc.)

• The ’cc-p’, stands for ’correlation consistent polarized’ and the ’V’ indicates they are valence only basis sets.

• They include successively larger shells of polarization (correlating) functions (𝑑, 𝑓, 𝑔, etc.).

• The prefix ’aug’ means that the basis is augmented with diffuse functions

• Examples: cc-pVTZ, aug-cc-pVDZ, aug-cc-pCV5Z

The complete basis set limit (CBS) can be approximately approached by extrapolation techniques

• Correlation consistent basis sets are built up by adding shells of functions to a core set of atomic Hartree-Fock functions.

• Each function in a shell contributes very similar amounts of correlation energy in an atomic calculation.

• For the 1st and 2nd row atoms, the cc-pVDZ basis set adds 1𝑠, 1𝑝, and 1𝑑 function. The cc-pVTZ set adds another 𝑠, 𝑝, 𝑑, and 𝑓 function, etc.

• For third-row atoms, additional functions are necessary; these are the cc-pV(n+d)Z basis sets.

****

C 0

S 8 1.00

8236.0000000 0.0005310

1235.0000000 0.0041080

280.8000000 0.0210870

79.2700000 0.0818530

25.5900000 0.2348170

8.9970000 0.4344010

3.3190000 0.3461290

0.3643000 -0.0089830

S 8 1.00

8236.0000000 -0.0001130

1235.0000000 -0.0008780

280.8000000 -0.0045400

79.2700000 -0.0181330

25.5900000 -0.0557600

8.9970000 -0.1268950

3.3190000 -0.1703520

0.3643000 0.5986840

S 1 1.00

0.9059000 1.0000000

S 1 1.00

0.1285000 1.0000000

P 3 1.00

18.7100000 0.0140310

4.1330000 0.0868660

1.2000000 0.2902160

P 1 1.00

0.3827000 1.0000000

P 1 1.00

0.1209000 1.0000000

D 1 1.00

1.0970000 1.0000000

D 1 1.00

0.3180000 1.0000000

F 1 1.00

0.7610000 1.0000000

****

****

C 0

S 8 1.00

6665.0000000 0.0006920

1000.0000000 0.0053290

228.0000000 0.0270770

64.7100000 0.1017180

21.0600000 0.2747400

7.4950000 0.4485640

2.7970000 0.2850740

0.5215000 0.0152040

S 8 1.00

6665.0000000 -0.0001460

1000.0000000 -0.0011540

228.0000000 -0.0057250

64.7100000 -0.0233120

21.0600000 -0.0639550

7.4950000 -0.1499810

2.7970000 -0.1272620

0.5215000 0.5445290

S 1 1.00

0.1596000 1.0000000

P 3 1.00

9.4390000 0.0381090

2.0020000 0.2094800

0.5456000 0.5085570

P 1 1.00

0.1517000 1.0000000

D 1 1.00

0.5500000 1.0000000

****

cc-pVDZ for carbon cc-pVTZ for carbon

Basis set superposition error

Calculations of interaction energies are susceptible to basis set superposition error (BSSE) if they use finite basis sets.

• As the atoms of interacting molecules or two molecules approach one another, their basis functions overlap. Each monomer ”borrows” functions from other nearby components, effectively increasing its basis set and improving the calculation.

• The counterpoise approach (CP) calculates the BSSE employing ”ghost orbitals”. In the uncorrected calculation of a dimer AB, the dimer basis set is the union of the two monomer basis sets. The uncorrected interaction energy is

𝑉AB 𝐺 = 𝐸AB 𝐺, 𝐴𝐵 − 𝐸A 𝐴 −𝐸B 𝐵

where 𝐺 denotes the coordinates that specify the geometry of the dimer and 𝐸AB 𝐺, 𝐴𝐵 the total energy of the dimer AB calculated with the full basis set 𝐴𝐵 of the dimer at that geometry. Similarly, 𝐸A 𝐴 and 𝐸B 𝐵 denote the total energies of the monomers A and B, each calculated with the appropriate monomer basis sets 𝐴 and 𝐵, respectively.

The counterpoise corrected interaction energy is

𝑉ABcc 𝐺 = 𝐸AB 𝐺, 𝐴𝐵 − 𝐸A 𝐺, 𝐴𝐵 −𝐸B 𝐺, 𝐴𝐵

where 𝐸A 𝐺, 𝐴𝐵 and 𝐸B 𝐺, 𝐴𝐵 denote the total energies of monomers A and B, respectively, computed with the dimer basis set at geometry 𝐺, i.e. in the calculation of monomer A the basis set of the ”other” monomer B is present at the same location as in dimer A, but the nuclei of B are not. In this way, the basis set for each monomer is extended by the functions of the other monomer.

The counterpoise corrected energy is thus

• The counterpoise correction provides only an estimate of the BSSE.

• BSSE is present in all molecular calculations involving finite basis sets but in practice its effect is important in calculations involving weakly bound complexes. Usually its magnitude is few kJ/mol to binding energies which is often very significant.

• The lowest-lying molecular orbitals are constrained to remain doubly-occupied in allconfigurations.

• The lowest-lying molecular orbitals are primarily these inner-shell atomic orbitals (or linear combinations thereof).

• The frozen core for atoms lithium to neon typically consists of the 1s atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals 1𝑠, 2𝑠, and 2𝑝.

• A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than are the valence electrons.

• The error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms.

• In fact, it is sometimes recommended that one employ the frozen core approximation as a general rule because most of the basis sets commonly used in quantum chemical calculations do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons.

The frozen core approximation

The pseudopotential is an attempt to replace the complicated effects of the motion of the core electrons and nucleus with an effective core potential (ECP)

Motivation:

• Reduction of basis set size

• Reduction of number of electrons

• Inclusion of relativistic and other effects

Approximations:

• Pseudopotentials imply the frozen core approximation

• Valence-only electrons

• Assumes that there is no significant overlap between coreand valence WF

The pseudopotentials

Towards exact solution of Scrödinger equation

Multi-configurational self-consistent field (MCSCF)

• MCSCF is a method to generate qualitatively correct reference states of molecules in cases where Hartree–Fock is not adequate

• It uses a linear combination of configuration state functions (CSF) or Slater determinants to approximate the exact electronic wavefunction

• It can be considered a combination between configuration interaction (where the molecular orbitals are not varied but the expansion of the wave function) and Hartree-Fock (where there is only one determinant but the molecular orbitals are varied).

MCSCF method is an attempt to generalize the Hartree–Fock model and to treat real chemical processes, where nondynamic correlation is important, while keeping the conceptual simplicity of the HF model as much as possible. Although MCSCF itself does not include dynamic correlations, it provides a good starting point for such studies.

Multi reference methods

Example: Hydrogen molecule MCSCF treatment

HF gives a reasonable description of H2 around the equilibrium geometry

• About 0.735 for the bond length compared to a 0.746 experimental value

• 84 kcal/mol for the bond energy (exp. 109 kcal/mol).

Problem: At large separations the presence of ionic terms H+ + H− (which have different energy than H + H) lead to an unphysical solution.

Solution: The total wave function of hydrogen molecule (including hydrogens A and B) can be written as a linear combination of configurations built from bonding and anti-bonding orbitals

𝜓MC = 𝐶1𝜓1 + 𝐶2𝜓2

where 𝜓1 is the bonding orbital 1𝑠A + 1𝑠B and 𝜓2 is the anti-bonding orbital 1𝑠A − 1𝑠B.

• In this multi configurational description of the H2 chemical bond, 𝐶1 = 1 and 𝐶2 = 0 close to equilibrium, and 𝐶1 will be comparable to 𝐶2 for large separations.

• In the complete active space CASSCF method the occupied orbital space is divided into a sets of active, inactive and secondary orbitals.The active space orbitals are highlighted in yellow, while inactive and secondary orbitals are greyed out.

• It allows complete distribution of active (valence) electrons in all possible ways. Corresponds to a FCI in the active space.

• The orbitals not incorporated in the active space remain either doubly occupied (inactive or core space) or empty (secondary).

• The restricted active space method (RASSCF) uses only selected subspaces of active orbitals.It could for example restrict thenumber of electrons to at most 2 in some subset.

For instance, the description of a double bond isomerization requires an active space including all π-electrons and π-orbitals of the conjugated system incorporating the reactive double bond. This choice is motivated by the need to allow for all possible variations in the overlap between the set of p-orbitals forming the reacting π-system along the reaction coordinate.

More generally, the selection of the active space electrons and orbitals is a “chemical problem”, and often is not a straightforward one.

It is often a challenge to generate configuration space sufficiently flexible to describe the physical process and yet sufficiently small to be computationally tractable.

Reaction schemes considered for the isomerization of the propene radical cation.

• Starts with a CASSCF calculation, which describes the static electron correlation by including the nearly degenerate electron configurations in thewavefunction.

• The dynamical electron correlation is included by substitutions of occupied orbitals by virtual orbitals in the individual configuration state functions (CSFs).

• The truncation of the expansion space to single and double substitutions is usually mandated by the very steep increase in the number of CSFs, and the consequent computational effort.

• The disadvantage of truncated CI is its lack of size-extensivity. A correction of this problem is by no means as straightforward in MRCI as in the single reference case.

A representation of the configurations included in the MRCI wavefunction taking one of the CASSCF configurations as the reference.

Multi reference configuration interaction (MRCI)

The multi-reference perturbation theory (MR-PT) is the most cost-effective multi-reference approach compared to multi-reference CI (MRCI) and multi-reference Coupled Cluster (MR-CC).

Example: In computational photochemistry and photobiology the CASPT2 method leads often to suitably accurate vertical excitation energies (with errors within 3 kcal/mol with respect to observation). This method has been shown to benefit from a balanced cancellation of errors. While the CASPT2 has been successfully applied in many studies, it typically requires experienced users that are familiar with its pitfalls. In fact, this requirement is generally true for multi-reference methods, such as CASSCF and MRCI, which require users capable of selecting variables such as the active space and the number of states to include in the calculation (i.e. they are not black-box methods).

MR-CC is approximately size-extensive but in general not clearly as advantageous compared to MR-CI as in single reference picture

The electron-electron distance 𝑟12 ought to be included into the wavefunctionif highly accurate computational results were to be obtained.

Example: For a helium atom we can choose the trial function as follows:

𝜙 𝑟1, 𝑟2, 𝑟12 = 𝐶 1 + 𝑝𝑟12 + 𝑞 𝑟1 − 𝑟22 exp −𝛼 𝑟1 + 𝑟2

where 𝐶 is normalization constant and 𝑝, 𝑞, and 𝛼 are variational parameters.

After optimization:

𝑝 = 0.30, 𝑞 = 0.13, and 𝛼 = 1.816

𝐸 = −2.9024 (Three parameters)

𝐸 = −2.9037462 (1024 parameters)

𝐸 = −2.9037843 (Experimental value)

Explicitly correlated methods

• R12 and F12 methods include two-electron basis functions that depend on the interparticledistance 𝑟12

• These theories bypass the slow convergence of conventional methods

• High accuracy can be achieved dramatically faster

• With these methods it has been achieved kJ/mol accuracy for molecular systems consisting of up to 18 atoms

• This result is well below the so called chemical accuracy, that is, an error of 1 kcal/mol (4.184 kJ/mol)

• Methods: CCSD-R12 or F12, CCSD(T)-R12 or F12, MP2-R12 or F12

Explicitly correlated methods

Example: Calculated and experimental geometric parameters

Molecule CCSD(T)-F12a/ VDZ-F12

CCSD(T)/ AVDZ CCSD(T)/ AV6Z Exp.

H2O

𝑅 0.9588 0.9665 0.9584 0.958

𝜃 104.36 103.94 104.45 104.5

H2S

𝑅 1.3367 1.3500 1.3378 1.336

𝜃 92.19 92.37 92.36 92.1

NH3

𝑅 1.0123 1.0237 1.0122 1.012

𝜃 106.59 105.93 106.62 106.7

• CCSD(T)-F12a is in impressive agreement with experiment

• Computational cost of CCSD(T)-F12a calculations are reduced because of negligible BSSE and small basis set

Computational expense can be reduced by simplification schemes.

• The local approximation. Interactions of distant pairs of localized molecular orbitals are neglected in the correlation calculation. This sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules. Methods employing this scheme are denoted by the prefix ”L”, e.g. LMP2. Scaling is reduced to N.

• The density fitting scheme The charge densities in the multi-center integrals are treated in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix ”df-”, for example the density fitting MP2 is df-MP2 (lower-case is advisable to prevent confusion with DFT).

• Both schemes can be employed together

For example, as in the recently developed df-LMP2 and df-LCCSD methods. In fact, df-LMP2 calculations are faster than df-Hartree–Fock calculations and thus are feasible in nearly all situations in which also DFT is. Compared to local methods (LMP2 and LCCSD) these methods are ten times faster.

Linear scaling approaches

Example: The interaction energy between two benzenes

Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.

• With this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the electron density.

• The multiple determinant calculations (CI for example) require very large basis sets due to the poor convergence of the correlation energy when the inter-electronic distance becomes very small. However, DFT can produce accurate results with relatively small basis sets.

• DFT has become the most popular and versatile method in computational chemistry, accounting for approximately 90% of all calculations today. The reason for this preference is that DFT scales with the same order as HF theory (𝑁3, where 𝑁 is proportional to system size)

Density functional theory

DFT avoids the expense of the more traditional methods, deriving the energy directly from the electron probability density, rather than the molecular wavefunction, thus drastically reducing the dimensionality of the problem. Regardless of how many electrons one has in the system, the density is always 3 dimensional.

• Some history:

DFT has been very popular for calculations in solid state physics since the 1970s. DFT calculations agreed quite satisfactorily with experimental data. Also, the computational costs were relatively low when compared to Hartree-Fock theory and its descendants. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. DFT is now a leading method for electronic structure calculations in chemistry and solid-state physics.

Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially:

• van der Waals forces (dispersion)

• charge transfer excitations

• transition states

• global potential energy surfaces and some other strongly correlated systems

• calculations of the band gap in semiconductors.

Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g., interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.

Electron density

• Define

𝜌1 𝐫 = 𝑁 … ψ 𝐫1, 𝐫2, … , 𝐫𝑁2 𝑑𝜎1𝑑𝐫2 … 𝑑𝐫𝑁

𝜌 𝐫 = 𝜌1 𝐫 𝑑𝜎1

o 𝜌1 describes the probability of finding any of the 𝑁 electrons within the volume element in the spin state 𝜎, with the other 𝑁-1 electrons having arbitrary positions and spin states

o 𝜌 is an observable (e.g. X-ray spectroscopy)

• Properties

𝜌 𝐫 → ∞ = 0

𝜌 𝐫 𝑑𝐫1 = 𝑁

Pair density

• Let’s generalize: the probability for finding two electrons in spin states and in the volume elements and is given by the pair density

𝜌2 𝐫1, 𝐫2 = 𝑁 𝑁 − 1 … ψ 𝐫1, 𝐫2, … , 𝐫𝑁2 𝑑𝜎1𝑑𝜎2𝑑𝐫3 … 𝑑𝐫𝑁

𝑃 𝐫1, 𝐫2 = 𝜌2 𝐫1, 𝐫2 𝑑𝜎1 𝑑𝜎2

• The pair density contains all information about electron correlation, and we can express the energy of any system in any state as

𝐸 = −1

2 ∇2𝜌 𝐫1 𝑑𝐫1 +

𝑖

𝜌 𝐫1

𝐫1 − 𝐑𝑖𝑑𝐫1 +

1

2

𝑃 𝐫1, 𝐫2

𝐫1 − 𝐫2𝑑𝐫1 𝑑𝐫2

• Rewrite

𝜌2 𝐫1, 𝐫2 = 𝜌1 𝐫1 𝜌1 𝐫2 1 + 𝑓 𝐫1, 𝐫2

Where 𝑓 𝐫1, 𝐫2 is called correlation factor. For example 𝑓 = 0 corresponds to uncorrelated case.

• The difference between the probability to find any electron in 𝑑𝐫2 while there is an electron in 𝑑𝐫1 and uncorrelated case is proportional to the exchange correlation hole ℎxc 𝐫1; 𝐫2

𝜌1 𝐫1 ℎxc 𝐫1; 𝐫2 = 𝜌2 𝐫1, 𝐫2 − 𝜌1 𝐫1 𝜌1 𝐫2 = 𝜌1 𝐫2 𝑓 𝐫1, 𝐫2

• We observe that

𝐸ee =1

2

𝑃 𝐫1, 𝐫2

𝐫1 − 𝐫2𝑑𝐫1 𝑑𝐫2 =

1

2

𝜌 𝐫1 𝜌 𝐫2

𝐫1 − 𝐫2𝑑𝐫1 𝑑𝐫2 +

1

2

𝜌 𝐫1 𝜌1 𝐫2 ℎxc 𝐫1; 𝐫2

𝐫1 − 𝐫2𝑑𝐫1 𝑑𝐫2

= 𝐽 𝜌 + 𝐸xc 𝜌

• The ℎxc can be formally splitted into

ℎxc = ℎx𝜎1=𝜎2 𝐫1; 𝐫2 + ℎc 𝐫1; 𝐫2

where ℎx is exchange hole, due to Pauli principle (wave function antisymmetry) and ℎc is correlationhole, due to electrostatic repulsion.

• Only the ℎxc can be given proper meaning

• The Hartree-Fock theory accounts for ℎx but neglects ℎc.

The Hohenberg-Kohn theorems

The Hohenberg-Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential

Theorem 1. The external potential and hence the total energy, is a unique functional of the electron density

• The first H-K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates.

• It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density.

• This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states.

Theorem 2. The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact ground state density

• The second Hohenberg-Kohn theorem has two drawbacks. Firstly, it assumes that there is no degeneracy in the ground state, and secondly the density has unknown form.

The uniform electron gas

There is no systematic way to find or improve a density functional. The most appealing way forward is to find the exact solution for a model system, and then assume that the system of interest behaves similarlyto the model.

• The first density functionals were due to Thomas, Fermi, and Dirac, all of which used the uniform electron gas as their model.

• The uniform electron gas is defined as a large number of electrons N in a cube of volume V, throughout which there is a uniform spread of positive charge sufficient to make the system neutral. The uniform gas is then defined as the limit N → ∞, V → ∞, with the density ρ = N/V remaining finite.

• Although it does bear some resemblance to electrons in metals, its widespread use is due to its simplicity: It is completely defined by one variable, the electron density ρ. Using the uniform electron gas, an expression for the kinetic energy (the Thomas-Fermi kinetic functional) can be derived

𝑇TF27 𝜌𝜎 =3

106𝜋2 2 3 𝜌𝜎

5 3 𝐫 𝑑𝐫, where σ can take the values of α or β.

The importance of simple Thomas-Fermi model is not how well it performs in computing the ground state energy and density but more as an illustration that the energy can be determined purely using the electron density.

When applied to atoms and molecules the Thomas-Fermi functional yields kinetic energies that are about 10% too small. Similarly, an expression for the exchange energy of the uniform electron gas can be calculated (the Dirac exchange functional)

𝐸xD30 𝜌𝜎 = −

3

2

3

4𝜋

1 3

𝜌𝜎 4 3 𝐫 𝑑𝐫

The Dirac functional also gives exchange energies that are roughly 10% smaller than those from HF theory.

The non-uniform electron gas

The electron densities of atoms and molecules are often far from uniform, so functionals based on systems which include an inhomogeneous density should perform better. When small ripples are placed on the uniform electron gas the kinetic energy becomes

𝑇W35 𝜌𝜎 = 𝑇TF27 𝜌𝜎 +1

8 𝜌𝜎

5 3 𝐫 𝑥𝜎2 𝐫 𝑑𝐫

where 𝑥 𝐫 is the reduced density gradient

𝑥 𝐫 =∇𝜌 𝐫

𝜌 4 3 𝐫

This functional is a large improvement on 𝑇TF27 𝜌𝜎 , yielding kinetic energies typically within 1% of HF theory. A similar correction was made to the Dirac exchange functional by Sham and Kleinman. The exchange energy becomes

𝐸xSK71 𝜌𝜎 = 𝐸x

D30 𝜌𝜎 −5

36𝜋 5 3 𝜌𝜎

4 3 𝐫 𝑥𝜎2 𝐫 𝑑𝐫

The corrected functional gives exchange energies that are typically within 3% of HF.

Kohn-Sham DFT

The kinetic energy has a large contribution to the total energy. Therefore even the 1% error in the kinetic energy of the Thomas-Fermi-Weizsacker model prevented DFT from being used as a quantitative predictive tool. Thus DFT was largely ignored until 1965 when Kohn and Sham introduced a method which treated the majority of the kinetic energy exactly.

Key idea: The intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The theory is based on the reference system: N noninteracting electrons moving in effective potential 𝑣eff, each in one of N orbitals, 𝜓𝑖.

The central equation in Kohn-Sham DFT is the one-electron Schrödingerlike equation expressed as:

−1

2∇𝑖

2 +𝑣eff 𝐫𝑖 𝜓𝑖 = 𝜖𝑖𝜓𝑖

The kinetic energy and electron density are given by

𝑇s 𝜌 =

𝑖

𝜓𝑖 −12 ∇𝑖

2 𝜓𝑖

𝜌 𝐫 =

𝑖

𝜓𝑖 𝐫 2

and the total energy is given by

𝐸 𝜌 = 𝑇s 𝜌 + 𝐸ee 𝜌 + 𝐸ne 𝜌 = 𝑇s 𝜌 + 𝐽 𝜌 + 𝐸xc 𝜌 + 𝐸ne 𝜌

Where 𝐸ne 𝜌 is energy arising from electron-nuclear interaction

The KS equations are very similar to the Hartree–Fock equations.

• Setting the exchange-correlation energy term to the HF exchange potential yields the HF equations.

• Just like the HF equations, the KS equations are solved iteratively.

Differences:

• The KS orbitals are simply a way of representing the density; they are not (as in HF) an approximation of the wavefunction.

• HF theory is variational, providing an upper bound to the exact energy, yet DFT is only variationalif the exact energy functional is used.

Because the KS equations so closely follow the restricted HF equations, both the restricted open shell and unrestricted methodologies are readily available. However, the KS equations are formally exact (given the exact 𝐸xc 𝜌 ), so it must be able to produce an excess of β electron density at points in the molecule, and therefore only the unrestricted formalism is appropriate.

• Just as in HF theory, the KS equations are solved by expanding the orbitals over a basis set.

• The major advantage of DFT is that the basis set requirements are far more modest than the more conventional correlated methods

• In DFT the basis set only needs to represent the one electron density – the inter-electron cusp is accounted for by the effective potential, 𝑣eff . In the more traditional methods the basis set describes the entire N-electron wavefunction, requiring an accurate description of the cusp which is sensitive to the basis set.

• The kinetic energy functional is known exactly.

• The exchange-correlation part of the total-energy functional remains unknown and must be approximated.

Local-density approximation

In local-density approximation (LDA), the exchange-correlation energy functional 𝐸xc 𝜌 depends only on the density at the coordinate where the functional is evaluated.

𝐸xcLDA 𝜌 = 𝜖xc 𝜌 𝜌 𝐫 𝑑𝐫

where 𝜖xc 𝜌 is the exchange-correlation energy density. The exchange-correlation energy is decomposed into exchange and correlation terms linearly:

𝐸xc = 𝐸x + 𝐸c

so that separate expressions for 𝐸x and 𝐸c are sought. The uniform electron gas functional is used for the 𝐸x:

𝐸x 𝜌𝜎 = −3

2

3

4𝜋

1 3

𝜌𝜎 4 3 𝐫 𝑑𝐫

The correlation energy is more complicated and numerous different approximations exist for 𝐸c.

• Strictly, the LDA is valid only for slowly varying densities.

• LDA works surprisingly well with calculations of atoms, molecules, and solids (especially for metals).

o Systematic error cancelation: Typically, in inhomogeneous systems LDA underestimates correlation but overestimates exchange, resulting in unexpectedly good energy value.

• LDA tends to overestimate cohesive energies by ∼15-20% and underestimates lattice constants by ∼2-3% for metals and insulators.

• Problem with LDA becomes more severe for weakly bonded systems, such as vdW and H-bonded systems.

o For example, the binding energy of the water dimer and the cohesive energy of bulk ice are both >50% too large with LDA compared to the experimental values.

o Long range vdW interactions are completely missing in LDA.

Generalized gradient approximation

LDA treats all systems as homogeneous. However, real systems are inhomogeneous. Generalized gradient approximation (GGA) takes this into account by including the derivative information of the density into the exchange-correlation functionals.

𝐸xcGGA 𝜌 = 𝑓 𝜌 𝐫 , ∇𝜌 𝐫 𝑑𝐫

• It is not the physics per se but obtained results that guide the mathematical constructs

o Some successful functionals are not based on any physical considerations

o For example let’s look two popular functionals: In PBE, the functional parameters are obtained from physical constraints (non-empirical). In B88, functional parameters are obtained from empirical fitting (empirical).

• GGAs are often called “semi-local” functionals due to their dependence on ∇𝜌 𝐫 .

• In comparison with LDA, GGA tend to improve total energies, atomization energies, energy barriers and structural energy differences. Especially for covalent bonds and weakly bonded systems many GGAs are far superior to LDA

o Overall though because of flexibility of a choice of 𝑓 𝜌 𝐫 , ∇𝜌 𝐫 a zoo of GGA functionals

have been developed and depending on the system under study a wide variety of results can be obtained.

• GGA expand and soften bonds, an effect that sometimes corrects and sometimes overcorrects the LDA prediction

• Whereas the 𝜖xc 𝜌 (in LDA) is well established, the best choice for 𝑓 𝜌 𝐫 , ∇𝜌 𝐫 is still a

matter of debate

The hybrid functionals

Q: Why bother with making GGA exchange functionals at all – we know that the HF exchange is exact; i.e.

These fourth generation functionals add “exact exchange”calculated from the HF functional to some conventional treatment of DFT exchange and correlation.

• LDA and GGA exchange and correlation functionals are mixed with a fraction of HF exchange

• The most widely used, particularly in the quantum chemistry community, is the B3LYP functional which employs three parameters, determined through fitting to experiment, to control the mixing of the HF exchange and density functional exchange and correlation.

Equilibrium C-C and C=C bond distances (Å)

Molecule HF B3LYP MP2 Exp.

But-1-yne-3-ene 1.439 1.424 1.429 1.431

Propyne 1.468 1.461 1.463 1.459

1,3-Butadiene 1.467 1.458 1.458 1.483

Propene 1.503 1.502 1.499 1.501

Cyclopropane 1.497 1.509 1.504 1.510

Propane 1.528 1.532 1.526 1.526

Cyclobutane 1.548 1.553 1.545 1.548

Cyclopropene 1.276 1.295 1.303 1.300

Allene 1.296 1.307 1.313 1.308

Propene 1.318 1.333 1.338 1.318

Cyclobutene 1.322 1.341 1.347 1.332

But-1-yne-ene 1.322 1.341 1.344 1.341

1,3-Butadiene 1.323 1.340 1.344 1.345

Cyclopentadiene 1.329 1.349 1.354 1.345

Mean error 0.011 0.006 0.007 -

The meta-GGAs

These are the third generation functionals and use the second derivative of the density,

∇2𝜌 𝐫 and/or kinetic energy densities, 𝜏 𝐫 =𝟏

𝟐 𝒊 𝜑 𝜌 𝟐, as additional degrees of freedom. In gas

phase studies of molecular properties meta-GGAs have been shown to offer improved performance over LDAs and GGAs.

Another class of functionals, known as hybrid meta-GGA functionals, is combination of meta-GGA and hybrid functionals with suitable parameters fitted to various molecular databases.

In practice, DFT can be applied in several distinct ways depending on what is being investigated.

• In solid state calculations, the local density approximations are still commonly used along with plane wave basis sets, as an electron gas approach is more appropriate for electrons delocalised through an infinite solid.

• In molecular calculations more sophisticated functionals are needed, and a huge variety of exchange-correlation functionals have been developed for chemical applications.

• In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part).

• Even more widely used is B3LYP which is a hybrid functional in which the exchange energy, in this case from Becke’s exchange functional, is combined with the exact energy from Hartree–Fock theory. The adjustable parameters in hybrid functionals are generally fitted to a ’training set’ of molecules.

• Unfortunately, although the results obtained with these functionals are usually sufficiently accurate for most applications, there is no systematic way of improving them (in contrast to methods like configuration interaction or coupled cluster theory)

• Hence in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.

DFT summary

Graphical models

In addition to numerical quantities (bond lengths and angles, energies, dipole moments,…) somechemically useful information is best displayed in the form of images. For example molecularorbitals, the electron density, electrostatic potential, etc. These objects can be displayed on screenusing isosurface

𝑓 𝑥, 𝑦, 𝑧 = constant

The constant may be some physical observable of interest, for example, the ”size” of the molecule.

Molecular Orbitals

Chemists are familiar with the molecular orbitals of simple molecules. They recognize the σ and π orbitals of acetylene, and readily associate these with the molecule’s σand π bonds

A simple example where the shape of the highest occupied molecular orbital (HOMO) “foretells” of chemistry is found in cyanide anion.

Cyanide acts as a nucleophile in SN2 reactions

:N≡C:− + CH3−I → :N≡C−CH3 + I−

The HOMO in cyanide is more concentrated on carbon (on the right) than on nitrogen suggesting, as is observed, that it will act as a carbon nucleophile.

Molecular orbitals do not even need to be occupied to be informative. For example, the lowest-unoccupied molecular orbital (LUMO) of perpendicularbenzyl cation anticipates the charge delocalization.

It is into the LUMO, the energetically most accessible unfilled molecular orbital, that any further electrons will go. Hence, it may be thought of as demarking the location of positive charge in a molecule.

Examination of the LUMO of methyl iodide helps to “rationalize”why iodine leaves following attack by cyanide.

This orbital is antibonding between carbon and iodine (there is a node in thebonding region), meaning that donation of the electron pair from cyanide will cause the CI bond to weaken and eventually break.

+

Woodward and Hoffmann first introduced organic chemists to the idea that so-called “frontier orbitals” (the HOMO and LUMO), which often provide the key to understanding why some chemical reactions proceed easily whereas others do not.

HOMO in cis-1,3-butadiene is able to interact favorably with the LUMO in ethylene (constructive overlap) to form cyclohexene

Interaction between the HOMO on one ethyleneand the LUMO on another ethylene is not favorable, and concerted addition to form cyclobutane would not be expected

+

+

-

-

-

+

+

+

+

+

-

-

-

-

Electron density

Isodensity surfaces may either serve to locate atoms, delineate chemical bonds, or to indicate overall molecular size and shape.

• The regions of highest electron density surround the heavy (non-hydrogen) atoms in a molecule. Thus, the X-ray diffraction experiment locates atoms by identifying regions of high electron density.

• Also interesting, are regions of slightly lower electron density. For example, isodensity surface(0.1 electrons/au3) for cyclohexanone conveys essentially the same information as a conventional skeletal structure model, that is, it depicts the locations of bonds

A low density surface (0.002 electrons/au3), serves to portray overall molecular size and shape. This is, of course, the same information portrayed by a conventional space-filling (CPK) model.

Bond surfaces (intermediate density) may be applied to elucidate bonding and not only to portray “known” bonding. For example, the bond surface for diborane clearly shows a molecule with very little electron density concentrated between the two borons.

This suggests that the appropriate Lewis structure is the one which lacks a boron-boron bond, rather than the one which has the two borons directly bonded.

Another important application of bond surfaces is to the description of the bonding in transition states. An example is the pyrolysis of ethyl formate, leading to formic acid and ethylene.

+

The bond surface offers clear evidence of a “late transition state”. The CO bond is nearly fully cleaved and the migrating hydrogen is more tightly bound to oxygen (as in the product) than to carbon (as in the reactant).

Spin density

For open-shell molecules, the spin density indicates the distribution of unpaired electrons. Spin density is an obvious indicator of reactivity of radicals (in which there is a single unpaired electron). Bonds will be made to centers for which the spin density is greatest. For example, the spin density isosurface for allyl radical suggests that reaction will occur on one of the terminal carbons and not on the central carbon.

Electrostatic potential

The value of the electrostatic potential (the energy of interaction of a positive point charge with the nuclei and electrons of a molecule) mapped onto an electron density isosurface may be employed to distinguish regions on the surface which are electron rich (“basic” or subject to electrophilic attack) from those which are electron poor (“acidic” or subject to nucleophilic attack).

• Negative potential surfaces serve to “outline” the location of the highest-energy electrons, for example lone pairs.

Example: A surface for which the electrostatic potential is negative, above and below the plane of the ring in benzene, and in the ring plane above the nitrogen in pyridine

benzene pyridine

While these two molecules are structurally very similar, potential surfaces make clear that this similarity does not carry over into their electrophilic reactivities.

Polarization potential

The polarization potential provides the energy due to electronic reorganization of the molecule as a result of its interaction with a point positive charge. For example, It properly orders the proton affinities (measure of gas-phase basicity, or energy released when molecule accept a proton) of trimethylamine, dimethyl etherand fluoromethane.

Local Ionization potential

The local ionization potential is intended to reflect the relative ease of electron removal (“ionization”) at any location around a molecule. For example, a surface of “low” local ionization potential for sulfur tetrafluoride demarks the areas which are most easily ionized.

Semiempirical methods

• Semiempirical methods of quantum chemistry start out from the ab initio formalism (HF-SCF) and then introduce assumptions to speed up the calculations, typically neglegting many of the less important terms in the ab initio equations.

• In order to compensate for the errors caused by these approximations, empirical parameters are incorporated into the formalism and calibrated against reliable experimental or theoretical reference data.

• It is generally recognized that ab initio methods (MP, CI, and CC) and even DFT can give the right result for the right reason, not only in principle, but often in practice, and that semiempirical calculations can offer qualitatively correct results of useful accuracy for many larger and chemically interesting systems.

• Semiempirical calculations are usually faster than DFT computations by more that two orders of magnitude, and therefore they often remain the method of choice in applications that involve really large molecules (biochemistry) or a large number of molecules or a large number of calculations (dynamics).

• Today, many chemical problems are solved by the combined use of ab initio, DFT, and semiempiricalmethods.

Basic concepts:

• A semiempirical model employs a Hartree-Fock SCF-MO treatment for the valence electrons with a minimal basis set.

• The core electrons are taken into a account through the effective nuclear charge, which is used in place of the actual nuclear charge to account for the electron-electron repulsions, or representedby ECP.

• Dynamic electron correlation effects are often included in an average sense.

• The standard HF equations are simplified by neglegting all three-center and four-center twoelectron integrals.

• One-center and two-center integral calculations are also simplified. For example CNDO (completeneglect of differential overlap), INDO (intermediate neglect of differential overlap), and NDDO (neglect of diatomic differential overlap) schemes differ how they introduce approximations in one-center and two-center integral calculations.

Consider the following two-electron integral

𝜓𝑖∗ 1 𝜓𝑗

∗ 21

𝑟12𝜓𝑗 2 𝜓𝑖 1 𝑑𝜏1𝑑𝜏2

where 𝜓 are expanded in terms of atom centered basis functions 𝜙 as usual

𝜓𝑖 =

𝑘

𝑐𝑖𝑘𝜙𝑘

Thus, the above integral includes terms of the following type

𝑘𝑙 𝑟𝑠 = 𝜙𝑘 1 𝜙𝑙 11

𝑟12𝜙𝑟 2 𝜙𝑠 2 𝑑𝜏1𝑑𝜏2

The zero differential overlap approximation ignores integrals that contain the products where 𝑘 is not equal to 𝑙 and 𝑟 is not equal to 𝑠

𝑘𝑙 𝑟𝑠 = 𝑘𝑘 𝑟𝑟

total number of such integrals is reduced approximately from 𝑁4 8 (Hartree Fock) to 𝑁2 2.

• The CNDO method use the zero differential overlap approximation completely.

Spherically symmetric orbitals only

• Methods based on the intermediate neglect of differential overlap, such as INDO, do not apply the zero differential overlap approximation when all four basis functions are on the same atom

One-centre repulsion integrals between different orbitals

• Methods that use the neglect of diatomic differential overlap, NDDO, do not apply the zero differential overlap approximation when the basis functions for the first electron are on the same atom and the basis functions for the second electron are on the same atom.

Includes some directionality of orbitals

• The approximations work reasonably well when the integrals that remain are parametrized

• The one-center and two-center integrals are determined directly from experimental data (one-center integrals derived from atomic spectroscopic data) or calculated using analytical formulas orrepresented by suitable parametric expressions (empirical or high-level ab initio).

• Most succesful semi-empirical methods (for studying ground-state potential energy surfaces) arebased on NDDO scheme

o In MNDO (Modified Neglect of Diatomic Overlap), the parametrisation is focused on ground state properties (heats of formation and geometries), ionization potentials, and dipole moments

o Later, MNDO was essentially replaced by two new methods, PM3 and AM1, which are similar but have different parametrisation methods (more parameters and thus, more flexibility).

Applications:

• Large biomolecules with thousands of atoms

o The accuracy of semiempirical methods is best for organic compounds

• Medicinal chemistry and drug design

o Semiempirical methods are well suited for quantitative structure-property relationship and quantitative structure-activity relationship (QSPR and QSAR, respectively) modeling.

• Nanoparticles

o Large fullerenes and nanotubes are prime examples

• Solids and surfaces

o Large clusters which approach bulk limit

• Direct reaction dynamics

o Thousands or even millions single point calculations

• Electronically excited states of large molecules and photochemistry. Alternative for TDDFT.

Molecular dynamics

• Molecular dynamics (MD) is computer simulation technique where the time evolutionof atoms is followed by solving their equations of motions.

• It uses a Maxwell-Botzmann averaging for thermodynamic properties

• Results emerge in a form of simulation. Changes in structures of systems, vibrations, as well as movements of particles are simulated.

• Simulation ”brings to life” the models yielding vast array of chemical and physicalinformation often surpassing (in content at least) the real experiments.

Molecular dynamics basics

• The laws of classical mechanics are followed, most notably Newton’s law:

F𝑖 = 𝑚𝑖a𝑖

for each atom 𝑖 in a system constituted by N atoms. Here, 𝑚𝑖 is the atom mass, a𝑖 = 𝑑2r𝑖 𝑑𝑡2 itsacceleration, and F𝑖 the force acting upon it.

• MD is a deterministic technique: given an initial set of positions and velocities, the subsequenttime evolution is in principle completely determined.

• In practice small numerical errors cause chaotic behaviour (butterfly effect).

• MD is a statistical mechanics method. It is a way to obtain a set of configurations or statesdistributed according to some statistical distribution function, or statistical ensemble.

• The properties, such as kinetic energy for example, are calculated using time averages. These are assumed to correspond to observable ensemble averages when the system is allowed to evolve in time indefinitely so system will eventually pass through all possible states (Ergodic hypothesis). Because the simulations are of fixed duration, one must be certain to sample a sufficient amount of phase space.

Phase Space

• For a system of N particles (e.g. atoms), the phase space is the 6N dimensional space of all the positions and momenta.

• At any given time, the state of the system (i.e. generally, the position and velocity of every atom) is given bya unique point in the phase space.

• The time evolution of the system can be seen as a displacement of the point in the phase space. Molecular dynamics as a simulation method is mainly a way of exploring, or sampling, the phase space.

• One of the biggest problem in molecular simulations is that the volume of the phase space (i.e., the number of accessible configurations for the system) is usually so huge that it is impossible to examine all of it.

• However, (for the case of a constant temperature system, usual in MD) different regions of the phase space have different probabilities to be observed.

• Boltzmann distribution says that the system has a higher probability to be in a low energy state. Molecular dynamics can be viewed as a way of producing configurations of the system (so, points in the phase space) according to their Boltzmann weight.

Modeling the system

• Choosing the potential energy function 𝑉(r1, … , r𝑁)

• Deriving the forces as the gradients of the potential with respect to atomic displacements:

F𝑖 = −∇r𝑖𝑉(r1, … , r𝑁)

• Writing the potential as a sum of pairwise interactions:

𝑉 r1, … , r𝑁 =

𝑖

𝑗>𝑖

𝜙 r𝑖 − r𝑗

• Introducing two-body interaction model, for example Lennard-Jones pair potential

𝜙LJ 𝑟 = 4𝜀𝜎

𝑟

12

−𝜎

𝑟

6

In practice it is customary to establish a cutoff radius 𝑅𝑐 so that the potential becomes

𝑉 𝑟 = 𝜙LJ 𝑟 − 𝜙LJ 𝑅𝑐 if 𝑟 ≤ 𝑅𝑐

0 if 𝑟 > 𝑅𝑐

Time integration

• The trajectories of interacting particles are calculated by integrating their equation of motion over time

• Time integration is based on finite difference methods, where time is discretized on a finite grid, the timestep ∆𝑡 being the distance between consecutive points in the grid.

• Knowing the positions and time derivatives at time 𝑡, the integration gives new quantities at a later time𝑡 + ∆𝑡.

• By iterating the procedure, the time evolution of the system can be followed for long times.

• The most commonly used time integration algorithm is propably the velocity Verlet algorithm, whereposition, velocities and accelerations at time 𝑡 + ∆𝑡 are obtained from the same quantities at time 𝑡 in thefollowing way

𝐫 𝑡 + ∆𝑡 = 𝐫 𝑡 + 𝐯 𝑡 ∆𝑡 + 1 2 𝐚 𝑡 ∆𝑡2

𝐯 𝑡 + ∆𝑡/2 = 𝐯 𝑡 + 1 2 𝐚 𝑡 ∆𝑡

𝐚 𝑡 + ∆𝑡 = − 1 𝑚 ∇𝑉(𝐫(𝑡 + ∆𝑡))

𝐯 𝑡 + ∆𝑡 = 𝐯 𝑡 + ∆𝑡/2 + 1 2 𝐚 𝑡 + ∆𝑡 ∆𝑡

• Velocities are required (to obtain kinetic energy 𝐾) to test the conservation of energy 𝐸 = 𝐾 + 𝑉.

Application Areas for MD

Materials Science

• Equilibrium thermodynamics

• Phase transitions

• Properties of lattice defects

• Nucleation and surface growth

• Heat/pressure processing

• Ion implantation

• Properties of nanostructures

Medicine

• Drug design and discovery

Chemistry

• Intra- and intermolecular interactions

• Chemical reactions

• Phase transitions

• Free energy calculations

Biophysics and biochemistry

• Protein folding and structure prediction

• Biocombatibility (cell wall penetration, chemical processes)

• Docking

Different levels of methods

• QM-Based Methods

o Limited to the range of hundreds of atoms

o Very short times in dynamical simulations (<100ps) on a supercomputer.

o Most reliable MD method applicable

o Can be used for treatment of chemical reactions

• Classical MD

o Whenever large systems, long time scales, and/or long series of events are needed for dynamical simulations, we must rely on classical (non-QM) MD

• Hybrid QM-MM

o QM and classical MD can be applied simultaneously

o System divided into parts

Examples:

A proton transfer dynamics of 2-aminopyridine dimer studied using ab initio molecular dynamics (QM-MD)(Phys. Chem. Chem. Phys., 2011, 13, 5881-5887)

Examples:

QM-MM MD was used to simulate microsolvation structure of Glycine in water

Glycine

Limitations of molecular dynamics

• Electrons are not present explicitly in classical MD

• Realism of potential energy surfaces

o Parameters are imperfect

• Classical description of atomic motions

o Quantum effects can become significant in any system as soon as T is sufficiently low

• The limitations on the size (number of atoms) and time of the simulation constrain the range of problems that can be addressed by the MD method

o The size of structural features of interest are limited to the size of the MD computational cell on the order of tens of nm.

o Using modern computers it is possible to calculate 106– 108 timesteps. Therefore MD can only simulate processes that occur within 1 – 100 ns. This is a serious limitation for many problems, for example thermally-activated processes.

o Record: Largest MD simulation

Periodic boundary conditions

• Particles are enclosed in a box

• Box is replicated to infinity by rigid translation to thecartesian directions

• All these replicated particles move together but only one of them is represented in a computer program

• Each particle in the box is not only interacting with otherparticles in the same box, but also with their images in other boxes

• In practice the number of interactions is limited becausecutoff radius of interaction potential

• Surface can be modeled by creating a slab

Running a simulation

• MD simulations are typically performed in the NVE (number of particles, volume, and energy remain constant), NVT and NPT ensembles

• During a simulation at constant energy the temperature will fluctuate. The temperature can be calculated from the velocities using the relationship:

𝐸𝑘 =3

2𝑁𝑘𝐵𝑇

𝑇 =1

3𝑁𝑘𝐵

𝑖=1

𝑁

𝑚𝑖𝑣𝑖2

Constant Temperature:

• Couple the system to an external heat bath that is fixed at the desired temperature

• The bath acts as a source of thermal energy, supplying or removing heat from the system as needed

Three essential steps taken to set up and run MD simulation:

1. Initialization

o Specify the initial coordinates and velocities

o Velocities are chosen randomly from Maxwell-Boltzmann or Gaussian distribution

2. Equilibrium

o Prior to starting a simulation it is advisable to do energy minimization

o Useful in correcting flaws such as steric clashes between atoms and distorted bond angles/lengths

3. Production

o Calculate thermodynamic properties of the system