Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto...

Daniel Sánchez PortalRicardo Díez Muiño

Centro de Física de MaterialesCentro Mixto CSIC-UPV/EHU

centro de física de materialesCFM

http://www.csic.es/

Electron correlation methods in Quantum Chemistry

Electronic structure calculations: Methodology and applications to nanostructures

Lectures on Quantum Chemistry:

Tuesday March 17th: 9.45 --> 12.30Theoretical background

Wednesday March 18th : 9.45 --> 12.30Practical exercise


Outline

• Brief introduction• Hartree- Fock• Basis sets• Configuration Interaction• Many-body perturbation theory• Coupled-cluster methods


Outline

• Brief introduction• Hartree- Fock• Basis sets• Configuration Interaction• Many-body perturbation theory• Coupled-cluster methods


PostHartree

Fock


Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry—an aberration which is happily almost impossible–it would occasion a rapid and widespread degeneration of that science.

Auguste Comte, 1830.


In conclusion, I would like to emphasize my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory, for increasingly many facets of chemical information, is already at hand. There is only one obstacle, namely, that someone must pay for the computing time.

Robert Mulliken. Nobel Prize address, 1966.


Quantum chemistry is quantum mechanics applied tothe electrons in atoms and molecules.

Used to determine:

1.- Structure of the molecule (bond lengths, angles)

2.- Electronic energy

(bond energies, enthalpies of formation, etc)

3.- Spectra (electronic, vibrational, rotational, etc)

4.- Electrical properties (dipole moment, polarizability)

5.- Molecular orbitals and derived properties

such as effective charges, bond orders.

6.- Barriers to reaction and other rate properties.

Goal of quantum chemistry methods:Multi-electron atoms and molecules

(r1,r2,...,rN )

1 2 1 2( , ,..., ) ( , ,..., )n N n NH r r r E r r r

21 1

2I

ii iI i jiI ij

ZH

r r


Wave function of many electrons in an external potential (Borh-Oppenheimer)

Finite system (no periodic boundary conditions)

Key quantity: The wave function• Fundamental object in quantum mechanics: wave function

• We want to find special wave functions such that

where

• This is a fundamentally many-body equation!

• A large variety of methods have been proposed and are being used to solve this problem.

(r1,r2,...,rN )

1 2 1 2( , ,..., ) ( , ,..., )n N n NH r r r E r r r

21 1

2I

ii iI i jiI ij

ZH

r r


Variational principle


This will be one of our main tools today.

It states that the energy calculated from an approximation to the true wavefunction will always be greater than the true energy:

Thus, the better the wavefunction, the lower the energy.

At a minimum, the first derivative of the energy will be zero.

Quantum Chemistry: Learning a new language


CIST, MP2, CC, CSF, TZV, ...

First (simple) approach: Hartree approximation

As a first guess, one may try to write the many-electron wavefunction as

a product of one-electron spin-orbitals ji(ri,si):


An important feature of the Hartree description is that the probability of finding one electron at a particular point in space is independent of the probability of finding any other electron at that point in space.

Thus, due to the independent particle model, the motion of the electrons in the Hartree approximation is uncorrelated.

Spin-orbitals


One-electron spin-orbitals ji(ri,si) are constructed as the product of a spatial orbital and a spin function. In general, they are molecular orbitals.

Hartree energy


Applying the variational principle to the above wave function, one can find the single-particle Hartree equations:

self-consistent equationsmean field

Antisymmetric wave function: Slater determinant


The main problem with Hartree’s wave function is that it violates Pauli´s principle. The wave function of fermions must be antisymmetric and therefore two fermions cannot be in the same quantum state.

Slater determinant

Antisymmetric wave function: Slater determinant


Slater determinant

-Exchanging any two rows of a determinant (exchanging two electrons) leads to a change in sign antisymmetry.

-Two electrons in the same quantum state two identical rows the determinant is zero.

Variational principle on a Slater determinant: Hartree-Fock


The minimization of the energy <Y|H|Y> assuming that the wave function Y is a Slater determinant leads to the Hartree-Fock approximation. The corresponding single-particle Hartree-Fock equations are the following:



Identical to Hartree New exchange term

Again, this is a mean field self-consistent model.

Coulomb term J Exchange term K



Notice that adding the term i=j in the sums modifies nothing: It cancels out.

Hartree-Fock equations: Fock operator


2 *2

2 2 21 '

2j

ext j j je

V e d e dm

x x xr x x x x x

r r r r

We can define the following operators by their action on an orbital:

xrr

xxxx jj dJ

2

x

rr

xxxxx

j

j dK*

and from them we define the Fock operator:

xxr KJVF ext2

2

1

so that the HF equations can be written as:

xx jjjF

Restricted Hartree-Fock (RHF) and Unrestricted Hartree-Fock (UHF)


RHF: the spatial part of the one-electron spin-orbitals ji(ri,si) is identical for spin-up and spin-down (closed-shell)

UHF: the spatial part of the one-electron spin-orbitals ji(ri,si) depend on the spin-orientation. Here, the wavefunction may be not a proper spin eigenfunction (spin contamination).

The energy of a UHF wave function is always lower than (or equal to) the corresponding RHF wave function (there is more flexibility in the former).

Accuracy of HF: Spin issues


Slater determinants are always eigenfunctions of Sz. However, they are not necessarily eigenfunctions of S2.

For the general case there are always linear combinations of determinants that are eigenfunctions of Sz and S2 at the same time.

Such spin-adapted linear combination of determinants (configurations) are needed to describe open-shell systems.

Accuracy of Hartree-Fock


Hartree-Fock calculations often account for ~99% of the total energy of the system.

The problem is that the remaining ~1% can determine the physical and chemical properties of the system.


Hence, we have to improve over HF: How to do that?

Accuracy of Hartree-Fock


Basis sets

Orbitals are usually expanded in basis sets.


Basis sets

Slater-type orbitals (STOs)

n,l,m (r,,) = Nn,l,m, Yl,m (,) rn-1 e-r

are characterized by quantum numbers n, l, and m and exponents (which characterize the radial 'size' ) .

Slater-type orbitals are similar to Hydrogenic orbitals in the regions close to the nuclei.

Specifically, they have a non-zero slope near the nucleus on which they are located

d/dr(exp(-r))r=0 = -

so they can have proper electron-nucleus cusps.


Basis sets

Cartesian Gaussian-type orbitals (GTOs)

a,b,c (r,,) = N'a,b,c, xa yb zc exp(-r2),

are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the

orbital, and exponents which govern the radial 'size’.

GTOs have zero slope near r=0 because

d/dr(exp(-r2))r=0 = 0.

The Coulomb cusp at the origin is not properly described. But, computationally, multi-center integrals are much more

efficiently obtained.


Basis sets

r

loose Gaussian

medium Gaussian

tight Gaussian

orbital with cusp at r = 0

To overcome the cusp weakness of GTO functions, it is common to combine two, three, or more GTOs, with combination coefficients that are fixed (and not treated as parameters), into new functions called contracted GTOs or CGTOs. However, it is not possible to correctly produce a cusp by combining any number of Gaussian functions because every Gaussian has a zero slope at r = 0 as shown here.


Basis sets: what to do with these building bricks

Minimum basis set: the number of basis functions is equal to the number of core and valence electrons in the atom.

Double zeta (DZ): there are twice as many basis functions as there are core and valence electrons.

Triple zeta (TZ): there are three times as many basis functions as the number of core and valence electrons.

Quadruple zeta (QZ), Pentuple Zeta (PZ or 5Z), etc.

In any of them: split valence basis means that only the number of basis functions representing the valence electrons is increased.

H C N

HCN molecule: DZ basis allows fordifferent bonding in different directions


Basis sets: what to do with these building bricks

Polarization functions: a basis function with a higher component of angular momentum is added, p-functions to s-based orbitals, d-functions to p-based orbitals, etc.

Double Polarization functions: basis functions with two higher components of angular momentum are added.

For instance, double zeta with polarization (DZP), triple zeta plus double polarization (TZDP), etc.

Polarization functions give angular flexibility in forming molecular orbitals between valence atomic orbitals.

Polarization functions also allow for angular correlations in describing the correlated motions of electrons.

H C N

Electron correlation


Hartree-Fock is an approximation: It replaces the instantaneous electron-electron repulsion by an average repulsion term.

Strictly speaking, electron correlation energy is defined as the difference between the HF energy and the lowest possible energy that one can obtain within a given basis set.

Physically, it corresponds to the fact that, on average, the electrons are further apart than the situation described by the (R)HF wave function.

A clear example in RHF: electrons are paired in molecular orbitals and the spatial overlap between the orbitals of such pair-electrons is exactly one!

Electron correlation methods: Post Hartree-Fock


To improve over Hartree-Fock and include electron-correlation, the easiest way is to start from the Hartree-Fock approximation and ADD new things.

Different methodologies will be defined by the different ways to ‘add’ things to Hartree-Fock. Typically, they fall into two classes:

• Wavefunction expansion: The most common approaches are Configuration Interaction (CI) and Coupled-Cluster Methods (CC, CCSD).

• Perturbation theory: The most common approach is Møller-Plesset (MP2 or MP4).

Configuration Interaction


CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants.




where we are adding new Slater determinants that are singly (Ys), doubly (Yd), triply (Yt), quadruply (Yq), etc. Excited relative to the original HF determinant.

These determinants are often referred to as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.



These determinants are often referred to as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.

Configuration Interaction (CI)



YCI=a0YHF+aSYS+aDYD+…=SaiYi

Again we use the variational principle and look for the ai

coefficients that make minimal the wave function energy.

Löwdin (1955): Complete CI gives exact wavefunction for the given atomic basis. For an infinite basis, it provides the exact solution.

Orbitals are NOT reoptimized in CI!




Brillouin’s theorem: Matrix elements between the HF reference determinant and singly excited states are zero.

Structure of the CI matrix




In order to develop a computationally tractable model, the number of excited determinants in the CI expansion must be reduced.

Truncating the expansion at one (Ys) does not improve the HF result because of Brillouin’s theorem.

The lowest CI level that improves over HF is CI with Doubles (CID).

The number of singles is much lower than the number of Doubles. Therefore, including singles is not a big deal: CI with Singles and Doubles (CISD).

Also with Triples: CISDT. Also with Quadruples (CISDTQ).




The lowest CI level that improves over HF is CI with Doubles (CID).

Weights of excited configuration in the Ne atom. Doubles have the highest weight!


Example: correlation in the H2 dissociation problem.

Let’s try to illustrate how CI accounts for electron correlation taking as an example the dissociation of

the hydrogen molecule H2

Take two 1s orbitals, one in each center of the molecule, cA and cB

cA cB

HF



The basis determinants for a full CI calculation are the following:

Double

Single

Single

Single : triplet SZ=1

Single : triplet SZ=-1

F2+F3 triplet SZ=0

F2-F3 singlet



The ground state F0 and the doubly excited F1 can be expanded in terms of the atomic orbitals:

ionic covalent

Now, if we increase the bond length towards infinity, the HF wave function is still a mixture of ionic and covalent components and, in the

dissociation limit will be 50% H+H- and 50% H0H0.

This is totally wrong!!

Electron correlation is missing: electrons try to avoid each other!



We can solve that by using full CI. The full CI matrix can be shown to be:

For 1Sg symmetry only these terms matter

The variational parameters allow us to choose the best combination for each bonding distance. For instance, the ionic component disappears for a1=-a0



The problem can also be treated with a UHF wave function. Although the UHF wave function does not solve everything: spin contamination.

We introduce a variational parameter c in the definition of the molecular orbitals. Now they are different for spin-up and spin-down.

ionic covalent

but now we have an additional

triplet component



All this is conspicuous in the energy diagram:


Multi-Reference calculations

For almost degenerate levels it is crucial to optimize the orbitals as well:

Multi-Reference Self-Consistent Field (MRSCF): a kind of CI in which the orbitals, as well as the coefficients, are optimized. Configurations included in MCSCF are defined by the active space.

Multi-Reference Configuration Interaction (MRCI): A MRSCF function is chosen as reference. Singles, doubles, etc., are generated out of all the determinants that enter the MRSCF.

Active space in CI


Reduced CI methods

Idea: Not all determinants are equally important.

Ansatz: Only allow excitations from a subset of orbitals into a subset of virtual orbitals (active space).Allow only a maximal number of excitations.


Many-body perturbation theoryIn perturbation theory, the Hamiltonian splits into:

Perturbation, i.e., its effect should be small!

Unperturbed Hamiltonian


Møller-Plesset Perturbation Theory

First, let us remember the Hartree-Fock equations:

2 *2

2 2 21 '

2j

ext j j je

V e d e dm

x x xr x x x x x

r r r r

We have the Fock operator, defined as:

These are the self-consistent equations that the single-particle wave functions should fulfill to obtain the minimum energy for a single-determinant many-body wave function.

xxr KJVF ext2

2

1

so that the HF equations can be written as:

xx jjjF



k k

M

kkext

M

kke VH

rrr

1

2

1

2

1

11

2

The exact Hamiltonian He is:

We define an unperturbed Hamiltonian H0 as the sum of Fock operators:

,11

20 2

1

kkk

M

kkext

M

kk KJVH xxr

M

kkFH

10



k k

M

kkext

M

kke VH

rrr

1

2

1

2

1

11

2



,11

20 2

1

kkk

M

kkext

M

kk KJVH xxr

M

kkFH

10

In H0 we are summing up twice the electron-electron Coulomb interaction



k k

M

kkext

M

kke VH

rrr

1

2

1

2

1

11

2



,11

20 2

1

kkk

M

kkext

M

kk KJVH xxr

M

kkFH

10

In H0 the electron-electron interaction is considered in an average way.



k k

M

kkext

M

kke VH

rrr

1

2

1

2

1

11

2



,11

20 2

1

kkk

M

kkext

M

kk KJVH xxr

M

kkFH

10

The perturbation is therefore H’=He-H0:

k

kkk k

KJH

xxrr

1

2

1Not that small!



k

kkk k

KJH

xxrr

1

2

1

,11

20 2

1

kkk

M

kkext

M

kk KJVH xxr

M

kkFH

10

The unperturbed Hamiltonian H0 is a sum of one-electron operators without coupling. Therefore the solution will be the sum of all energies of the independent systems and the wavefunction will be the (antisymmetrized) product of the one electron wavefunctions (orbitals).

In other words, the ground state wave function will be the Hartree-Fock wavefunction Y0

HF.

The unperturbed energy is E0=<Y0HF | H0 |Y0

HF > ≠ EHF

The energy adding the first-order correction is E1=<Y0HF | H0+ H’ |Y0

HF > = EHF!!

(it is the matrix element that we have varied to obtain the HF equations)


Møller-Plesset Perturbation Theory (MP)

excitedii

i

EE

HMPE

0

2

002 )2(

Electron correlation corrections start at second order with this choice of the unperturbed Hamiltonian.

MP2 means second-order in Moller-Plesset expansion.

The perturbative correction is:

where the excited states Y(i)0. are eigenstates of the unperturbed Hamiltonian H0,

i.e., they are determinants in which excitations have been created.

Actually, doubly-excited (not single-excited) determinants are the first contribution!

Following contributions to the perturbative expansion are MP3, MP4, etc.


Convergence of MP perturbative series

No smooth convergence, or no convergence at all is possible!!

In practice, only low orders of the expansion can be included.


Coupled-cluster methodsThe basic ansatz of coupled cluster theory is that the exact many-electron wavefunction Y may be generated by the operation of an exponential operator on a single determinant.

F0 is a single determinant wave function (usually, the Hartree-Fock wave function F0 = YHF is used)

T is an excitation operator. The excitation operator can be written as a linear combination of single, double, triple, etc excitations, up to N fold excitations for an N electron system:


Coupled-cluster methods

etc.

and the action of each Ti operator is to create the full set of i-excitations

there is only one way to have a single excitation T1, but two ways to generate double excitations: a double excitation (T2) and two consecutive single excitations (T1 T1).



etc.

and the action of each Ti operator is to create the full set of i-excitations

there is only one way to have a single excitation T1, but two ways to generate double excitations: a double excitation (T2) and two consecutive single excitations (T1 T1).

To construct the coupled cluster wavefunction one must then determine

the various amplitudes t through a system of coupled equations.


Coupled-cluster (CC) methods… in practice

In coupled cluster methods, for a given type of corrections (say, single excitations, for instance), all required terms are included.

As a consequence, the method scales like M2N+2 for M basis functions and N electrons a very expensive scaling!!

In practice, only corrections up to a given term are included:

If higher-order terms are calculated in perturbation theory, they are indicated with Parentheses. For instance CCSD(T) means that the triples are obtained perturbatively.


Coupled-cluster (CC) methods… in practice

In coupled cluster methods, for a given type of corrections (say, single excitations, for instance), all required terms are included.

As a consequence, the method scales like M2N+2 for M basis functions and N electrons a very expensive scaling!!

In practice, only corrections up to a given term are included:

Almost the only one used in practice.

In summary:

• Quantum chemistry methods are very accurate but computationally expensive.

• Key points: choice of basis set and methodology used to improve over Hartree-Fock.


Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto...

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