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Many-Body Perturbation Theory

 Overview The energies and properties from the Hartree-Fock method can be improved by using Rayleigh-Schrödinger perturbation theory.  In this lesson we explore the theory and use of a form of Rayleigh-Schrödinger perturbation theory for many-particle quantum systems known as second-order many-body perturbation theory (MP2).  The Hamiltonian operator in MP2 is expressed as a reference Hamiltonian operator and a perturbation operator H(l) = H0 + l V where H0 = F and V = H - F in which F is the Fock operator whose eigenvalues are the molecular orbital energies of the Hartree-Fock method.  When the perturbation parameter equals one (l = 1), the Hamiltonian operator H(1) becomes equal to the exact Hamiltonian operator in the selected orbital basis set.  Computing the energy through second order in the perturbation parameter yields the MP2 method.  MP2 is one of the simplest methods to approximate the correlation energy, the energy difference between the energy from solution of the Schrödinger equation and the energy from Hartree-Fock method in the selected orbital basis set.

MP2

After loading the Quantum Chemistry package, we explore the equilibrium bond length, dipole moment and potential energy curve of carbon monoxide with comparisons to experiment.

Quantum Chemistry

We set the number of Digits to be used in computations to 15 and load the Quantum Chemistry package using Maple's with command.

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 ${\mathrm{Digits}}{≔}{15}$ (2.1.1)
 > $\mathrm{with}\left(\mathrm{QuantumChemistry}\right);$
 $\left[{\mathrm{AOLabels}}{,}{\mathrm{ActiveSpaceCI}}{,}{\mathrm{ActiveSpaceSCF}}{,}{\mathrm{AtomicData}}{,}{\mathrm{BondAngles}}{,}{\mathrm{BondDistances}}{,}{\mathrm{Charges}}{,}{\mathrm{ChargesPlot}}{,}{\mathrm{CorrelationEnergy}}{,}{\mathrm{CoupledCluster}}{,}{\mathrm{DensityFunctional}}{,}{\mathrm{DensityPlot3D}}{,}{\mathrm{Dipole}}{,}{\mathrm{DipolePlot}}{,}{\mathrm{Energy}}{,}{\mathrm{ExcitationEnergies}}{,}{\mathrm{ExcitationSpectra}}{,}{\mathrm{ExcitationSpectraPlot}}{,}{\mathrm{ExcitedStateEnergies}}{,}{\mathrm{ExcitedStateSpins}}{,}{\mathrm{FullCI}}{,}{\mathrm{GeometryOptimization}}{,}{\mathrm{HartreeFock}}{,}{\mathrm{Interactive}}{,}{\mathrm{Isotopes}}{,}{\mathrm{MOCoefficients}}{,}{\mathrm{MODiagram}}{,}{\mathrm{MOEnergies}}{,}{\mathrm{MOIntegrals}}{,}{\mathrm{MOOccupations}}{,}{\mathrm{MOOccupationsPlot}}{,}{\mathrm{MOSymmetries}}{,}{\mathrm{MP2}}{,}{\mathrm{MolecularData}}{,}{\mathrm{MolecularGeometry}}{,}{\mathrm{NuclearEnergy}}{,}{\mathrm{NuclearGradient}}{,}{\mathrm{OscillatorStrengths}}{,}{\mathrm{Parametric2RDM}}{,}{\mathrm{PlotMolecule}}{,}{\mathrm{Populations}}{,}{\mathrm{RDM1}}{,}{\mathrm{RDM2}}{,}{\mathrm{RTM1}}{,}{\mathrm{ReadXYZ}}{,}{\mathrm{Restore}}{,}{\mathrm{Save}}{,}{\mathrm{SaveXYZ}}{,}{\mathrm{SearchBasisSets}}{,}{\mathrm{SearchFunctionals}}{,}{\mathrm{SkeletalStructure}}{,}{\mathrm{Thermodynamics}}{,}{\mathrm{TransitionDipolePlot}}{,}{\mathrm{TransitionDipoles}}{,}{\mathrm{TransitionOrbitalPlot}}{,}{\mathrm{TransitionOrbitals}}{,}{\mathrm{Variational2RDM}}{,}{\mathrm{VibrationalModeAnimation}}{,}{\mathrm{VibrationalModes}}{,}{\mathrm{Video}}\right]$ (2.1.2)
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Carbon Monoxide

We compute the equilibrium bond length and the dipole moment of carbon monoxide.

Equilibrium Bond Length

To compute the equilibrium bond length, we select a set of bond distances from the roots of the sixth-order Chebyshev polynomial that are suitable for interpolation

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 ${\mathrm{bond_distances}}{≔}\left[{1.01340742}{,}{1.03928932}{,}{1.08411810}{,}{1.13588190}{,}{1.18071068}{,}{1.20659258}\right]$ (2.2.1.1)

We define a list of molecular geometries with each geometry corresponding to one of the bond distances

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 ${\mathrm{molecules}}{≔}\left[\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.01340742}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.03928932}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.08411810}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.13588190}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.18071068}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.20659258}\right]\right]\right]$ (2.2.1.2)

The energies for each geometry may be then readily computed with the Energy command in the Quantum Chemistry package in which we set method = MP2.

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 ${\mathrm{energies}}{≔}\left[{-113.21146174}{,}{-113.22841767}{,}{-113.24555326}{,}{-113.25064671}{,}{-113.24594169}{,}{-113.24039923}\right]$ (2.2.1.3)

We use polynomial interpolation to generate a polynomial in the bond distance R

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 ${\mathrm{pes}}{≔}{-}{9.66039475}{}{{R}}^{{5}}{+}{62.38556861}{}{{R}}^{{4}}{-}{163.80401804}{}{{R}}^{{3}}{+}{218.86658796}{}{{R}}^{{2}}{-}{148.46546961}{}{R}{-}{72.52251752}$ (2.2.1.4)

The potential energy surface (curve) can be plotted

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Finally, we differential the potential energy curve with respect to R and set the derivative to zero.

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 ${\mathrm{eq}}{≔}{-}{48.30197374}{}{{R}}^{{4}}{+}{249.54227445}{}{{R}}^{{3}}{-}{491.41205412}{}{{R}}^{{2}}{+}{437.73317592}{}{R}{-}{148.46546961}{=}{0}$ (2.2.1.5)

Solving the resulting equation yields the equilibrium bond length

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 ${\mathrm{R_eq}}{≔}{1.13054421}$ (2.2.1.6)
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(a) By changing the basis keyword in the Energy command, repeat the above computations for the following larger basis sets: (i) cc-pVTZ, (ii) cc-pVQZ, and (iii) cc-pV5Z, and report your results as a table.  (Note that the basis sets are increasing in size as cc-pVDZ < cc-pVTZ < cc-pVQZ < cc-pV5Z)

(b) Does the bond length of CO from the MP2 method increase or decrease with larger basis sets?

(c) How does the computed bond length of CO compare with experiment?  (Hint: Check the NIST web site https://cccbdb.nist.gov/expgeom2x.asp?casno=630080)

Dipole Moment

Let us compute the electric dipole moment of CO from MP2 at the equilibrium bond length in the cc-pVDZ basis set.  First, we define the molecule's geometry as a Maple list of lists giving the atom's names and xyz coordinates

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 ${\mathrm{molecule}}{≔}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.11014350}\right]\right]$ (2.2.2.1)

Using the Dipole command, we can compute the dipole from the MP2 method in the cc-pVDZ basis set

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 $\left[\begin{array}{cc}{"X"}& 0.000000000\\ {"Y"}& 0.000000000\\ {"Z"}& -0.0127400169\end{array}\right]$ (2.2.2.2)

For comparison we can also compute the dipole from the Hartree-Fock method

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 $\left[\begin{array}{cc}{"X"}& 0.000000000\\ {"Y"}& 0.000000000\\ {"Z"}& -0.145818057\end{array}\right]$ (2.2.2.3)

(d) By changing the basis keyword in the Energy command, repeat the above computations for the following larger basis sets: (i) cc-pVTZ, (ii) cc-pVQZ, and (iii) cc-pV5Z, and report your results as a table.

(e) Does the dipole moment of CO from the MP2 method increase or decrease with larger basis sets?  Does it appear to converge in the large-basis-set limit?

(f) Is the dipole moment of CO from either Hartree-Fock or MP2 in agreement with the experimental value available at NIST (https://cccbdb.nist.gov/diplistx.asp)?  Which is closer?

Potential Energy Curve

To compute the potential energy curve, we select a set of bond distances from the roots of the sixth-order Chebyshev polynomial that are suitable for interpolation

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 ${\mathrm{bond_distances}}{≔}\left[{0.98180834}{,}{1.13405483}{,}{1.39775350}{,}{1.70224650}{,}{1.96594517}{,}{2.11819166}\right]$ (2.2.3.1)

We define a list of molecular geometries with each geometry corresponding to one of the bond distances

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 ${\mathrm{molecules}}{≔}\left[\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{0.98180834}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.13405483}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.39775350}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.70224650}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{1.96594517}\right]\right]{,}\left[\left[{"C"}{,}{0}{,}{0}{,}{0}\right]{,}\left[{"O"}{,}{0}{,}{0}{,}{2.11819166}\right]\right]\right]$ (2.2.3.2)

The energies for each geometry may be then readily computed with the Energy command in the Quantum Chemistry package.

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 ${\mathrm{energies_mp2}}{≔}\left[{-112.95544897}{,}{-113.04050587}{,}{-112.96547532}{,}{-112.80294666}{,}{-112.60289471}{,}{-112.54447209}\right]$ (2.2.3.3)

We use polynomial interpolation to generate a polynomial in the bond distance R

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 ${\mathrm{pes_mp2}}{≔}{-}{3.48933102}{}{{R}}^{{5}}{+}{27.24786757}{}{{R}}^{{4}}{-}{84.47258157}{}{{R}}^{{3}}{+}{130.33469869}{}{{R}}^{{2}}{-}{99.84887326}{}{R}{-}{82.74826166}$ (2.2.3.4)

Similarly, for comparison we can compute the energies with the Hartree-Fock method

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 ${\mathrm{energies_hf}}{≔}\left[{-112.68907086}{,}{-112.74866568}{,}{-112.62428645}{,}{-112.43823734}{,}{-112.34119827}{,}{-112.31450377}\right]$ (2.2.3.5)

We use polynomial interpolation to generate a polynomial in the bond distance R

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 ${\mathrm{pes_hf}}{≔}{-}{0.86844042}{}{{R}}^{{5}}{+}{7.99580630}{}{{R}}^{{4}}{-}{29.34827331}{}{{R}}^{{3}}{+}{53.21006552}{}{{R}}^{{2}}{-}{46.83555798}{}{R}{-}{96.85908538}$ (2.2.3.6)

The potential energy curves from the Hartree-Fock method (red) and the MP2 method (blue) can be plotted together

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(g) Does the MP2 method improve upon the Hartree-Fock method around the equilibrium bond length?

(h) Explain what MP2 predicts upon dissociation.  Is this behavior physical or an artifact of the second-order perturbation theory?

References

 1 C. Møller and M. S. Plesset, Phys. Rev. 46, 618–622 (1934). "Note on an Approximation Treatment for Many-Electron Systems"
 2 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover Books, New York, 1996).