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Lecture 4. Perturbation theory September 22 – 1 / 17

Introduction to Computational Chemistry

Vesa HanninenLaboratory of Physical Chemistry

Chemicum 4th floor

[email protected]

September 20, 2012



Perturbation theory


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

The starting point is eigenfunction of the Fock-operator.

It improves on the Hartree–Fock method by adding electron correlationeffects.

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

ground state energy.MP methods (MP2, MP3, MP4, ...) are implemented in many computationalchemistry codes. Higher level MP calculations, generally only MP5, arepossible in some codes. However, they are rarely used because of their costs.



Quantum mechanical perturbation theory


The Scrodinger equation for the perturbed state Ψn is

H Ψn = ( H (0) + λV )Ψn = E nΨn (1)

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 discretestates.

The expressions produced by perturbation theory are not exact

Accurate results can be obtained as long as the expansion parameter λis very small.

We expand Ψn and E n in Taylor series in powers of λ. The eigenvalueequation becomes

( H (0) + λV )(i=0

λiΨ(i)n ) = (i=0

λiE (i)n )(i=0

λiΨ(i)n ) (2)





Writing only the first terms we obtain

( H (0) + λV )(Ψ(0)n + λΨ(1)

n ) = (E (0)n + λE (1)n )(Ψ(0)

n + λΨ(1)n ) (3)

The zeroth-order equation is simply the Schrodinger equation for the

unperturbed systemH

(0)Ψ(0)n = E

(0)n Ψ(0)

n (4)

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

H (0)Ψ(1)

n + V Ψ(0)n = E

(0)n Ψ(1)

n + E (1)n Ψ(0)

n (5)

When this is multiplied through by Ψ(0)∗n and integrated, the first term on the

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

E (1)n =

Ψ(0)nV Ψ(0)n

(6)

This is simply the expectation value of the perturbation Hamiltonian while thesystem is in the unperturbed state.





Interpretation of the first order correction to energy:

E (1)n =

Ψ(0)n

V Ψ(0)

n

(7)

The perturbation is applied, but we keep the system in the quantumstate |Ψ(0)

n , which is a valid quantum state though no longer an energyeigenstate.

The perturbation causes the average energy of this state to increase byΨ

(0)n |V |Ψ

(0)n .

The true energy shift is slightly different, because the perturbedeigenstate is not exactly the same as |Ψ

(0)n .

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 theexpression derived earlier

H (0)Ψ(1)

n + V Ψ(0)n = E

(0)n Ψ(1)

n + E (1)n Ψ(0)

n (8)

and multiply it by Ψ(0)∗m , (m = n) from left and integrate. We obtain

(E (0)m −E (0)n )Ψ(0)

m |Ψ(1)n = −Ψ(0)

m |V |Ψ(0)n (9)

We expand |Ψ(1)n as

|Ψ(1)n =m

am|Ψ(0)m (10)

When two above equations are combined, we obtain

|Ψ(1)n =

m=n

Ψ(0)m |V |Ψ

(0)n

E (0)n −E

(0)m

|Ψ(0)m (11)

The first-order change in the n-th energy eigenfunction has a contributionfrom each of the energy eigenstates m = n.





The second order correction to the energy is

E (2)n =

m=n

|Ψ(0)m |V |Ψ

(0)n |

2

E (0)n −E

(0)m

(12)

Conclusion:

Each term is proportional to the matrix element Ψ(0)m |V |Ψ

(0)n

This is a measure of how much the perturbation mixes eigenstate n

with eigenstate m

It is also inversely proportional to the energy difference betweeneigenstates m and n, which means that the perturbation deforms theeigenstate to a greater extent if there are more eigenstates at nearby

energies.

Expression is singular if any of these states have the same energy asstate n, which is why we assumed that there is no degeneracy

Higher-order deviations can be found by a similar procedure



MP perturbation theory


The MP-energy corrections are obtained with the perturbation (correlationpotential):

V = H − F − Φ0|H − F |Φ0 (13)

where H is the usual electronic Hamiltonian and the Slater determinant Φ0 is

the eigenfunction of the Fock-operator F

F Φ0 = 2(

N/2

i=1

ǫi)Φ0 (14)

where N is the number of electrons of the molecule under consideration andǫi is the orbital energy belonging to the doubly occupied spatial orbital. Theshifted Fock operator

H 0 = F + Φ0|H − F |Φ0 (15)

serves as unperturbed (zeroth-order) operator. The Slater determinant Φ0

being an eigenfunction of F , it follows that

H 0Φ0 = Φ0|H |Φ0Φ0 (16)





so that the zeroth-order energy is the expectation value of H with respect toΦ0, i.e., the Hartree–Fock energy:

E MP 0 = E HF = Φ0|H |Φ0 (17)

Since the first-order MP energy

E MP 1 = Φ0|V |Φ0 = 0 (18)

is zero, the lowest-order MP correlation energy appears in second order. Thisresult is the Møller–Plesset theorem: The correlation potential does not

contribute in first-order to the exact electronic energy.





In order to obtain the MP2 formula for a closed-shell molecule, the secondorder formula is written on basis of doubly-excited Slater determinants(singly-excited Slater determinants do not contribute, see Brillouin theorem).After integrating out spin, it becomesE MP 2 =

i,j,a,b

ψi(1)ψj(2)| 1

r12|ψa(1)ψb(2)×

2ψa(1)ψb(2)| 1r12 |ψi(1)ψj(2) − ψa(1)ψb(2)| 1r12 |ψj(1)ψi(2)

ǫi + ǫj − ǫa − ǫb(19)

where ψi and ψj are occupied orbitals and ψa and ψb are virtual (unoccupied)orbitals. The quantities ǫi, ǫj , ǫa, and ǫb are the corresponding orbitalenergies. Through second-order in the correlation potential, the totalelectronic energy is given by the Hartree–Fock energy plus second-order MPcorrection:

E = E HF + E MP 2 (20)





Concluding remarks

Systematic studies of MP perturbation theory have shown that it is notnecessarily a convergent theory at high orders. The convergenceproperties 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 MP4level are in no way better than their MP2 counterparts, even for smallmolecules.

For open shell molecules, MPn-theory can directly be applied only tounrestricted Hartree–Fock reference functions. However, the resultingenergies often suffer from severe spin contamination, leading to verywrong results. A much better alternative is to use one of the MP2-like

methods based on restricted open-shell Hartree–Fock references.



Example: Atomization energies


Table 1: Calculated and experimental electronic atomization energies (kJ/mol)Molecule HF CCSD MP2 Exp.

F2 -155.3 128.0 185.4 163.4

H2 350.8 458.1 440.7 458.0HF 405.7 583.9 613.8 593.2H2O 652.3 960.2 996.1 975.3O3 -238.2 496.1 726.6 616.2

CO2 1033.4 1573.6 1745.2 1632.5

C2H4 1793.9 2328.9 2379.3 2359.8CH4 1374.1 1747.0 1753.1 1759.3

Accuracy of the MP2 is satisfactory despite its relatively low

computational cost MP2 usually overestimates bond energies



Example: Reaction enthalpies


Table 2: Calculated and experimental electronic reaction enthalpies (kJ/mol)Reaction CCSD MP2 Experiment

CO + H2 → CH2O -23.4 -25.0 -21.8

H2O + F2 → HOF + HF -123.3 -127.2 -129.4N2 + 3H2 → 2NH3 -173.1 -164.4 -165.4C2H2 + H2 → C2H4 -209.7 -196.1 -203.9

CO2 + 4H2 → CH4 + 2H2O -261.3 -237.3 -245.32CH2 → C2H4 -830.1 -897.7.9 -845.7

O3 + 3H2 → 3H2O -1010.1 -939.7 -935.5

The accuracy of MP2 is in the same level than CCSD

It is problematic to improve MPPT calculations systematically



Review: Coupled Cluster method


Coupled cluster (CC) method, especially The CCSD(T), has becomethe ”gold-standard of quantum chemistry”.

The computatinal cost is very high. So, in practice, it is limited torelatively small systems.

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

Some of the most accurate calculations for small to medium sized

molecules use this method.

A drawback of the method is that it is not variational

The abbreviations for coupled-cluster methods begin with the letters

CC followed by S - for single, D - for double excitations, etc.



Hybrid coupled cluster CCSD(T)


T in the brackets means perturbative triple excitations

Is the most popular hybrid CC method

For high precision work, the CCSD model is usually not accurate and

CCSDT model is too expensive

Example: Water molecule CCSD(T) calculations

For OH distances less than 3.5 A, the CCSD(T) works well, giving

about 90% of the full CCSDT triples correction.

The model breaks down at larger distances

The unrestricted CCSD(T) (based on UHF reference) does not providegood description of the dissociation process.



Example: Atomization energies


Table 3: Calculated and experimental electronic atomization energies (kJ/mol)Molecule HF CCSD CCSD(T) Exp.

F2 -155.3 128.0 161.1 163.4

H2 350.8 458.1 458.1 458.0HF 405.7 583.9 593.3 593.2H2O 652.3 960.2 975.5 975.3O3 -238.2 496.1 605.5 616.2

CO2 1033.4 1573.6 1633.2 1632.5

C2H4 1793.9 2328.9 2360.8 2359.8CH4 1374.1 1747.0 1759.4 1759.3

CCSD(T) calculations produce accurate results

Only for most problematic systems (such as ozone) higher ordercorrections are desirable



Example: Reaction enthalpies


Table 4: Calculated and experimental electronic reaction enthalpies (kJ/mol)Reaction CCSD CCSD(T) Experiment

CO + H2 → CH2O -23.4 -23.0 -21.8

H2O + F2 → HOF + HF -123.3 -119.5 -129.4N2 + 3H2 → 2NH3 -173.1 -165.5 -165.4C2H2 + H2 → C2H4 -209.7 -205.6 -203.9

CO2 + 4H2 → CH4 + 2H2O -261.3 -244.7 -245.32CH2 → C2H4 -830.1 -844.9 -845.7

O3 + 3H2 → 3H2O -1010.1 -946.6 -935.5

CCSD(T) is generally an improvement over CCSD

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

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