Many-Body Perturbation Theory
Copyright (c) RDMCHEM LLC 2020
Overview
MP2
References
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.
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.
Digits ≔ 15;
Digits≔15
withQuantumChemistry;
AOLabels,ActiveSpaceCI,ActiveSpaceSCF,AtomicData,BondAngles,BondDistances,Charges,ChargesPlot,CorrelationEnergy,CoupledCluster,DensityFunctional,DensityPlot3D,Dipole,DipolePlot,Energy,ExcitationEnergies,ExcitationSpectra,ExcitationSpectraPlot,ExcitedStateEnergies,ExcitedStateSpins,FullCI,GeometryOptimization,HartreeFock,Interactive,Isotopes,MOCoefficients,MODiagram,MOEnergies,MOIntegrals,MOOccupations,MOOccupationsPlot,MOSymmetries,MP2,MolecularData,MolecularGeometry,NuclearEnergy,NuclearGradient,OscillatorStrengths,Parametric2RDM,PlotMolecule,Populations,RDM1,RDM2,RTM1,ReadXYZ,Restore,Save,SaveXYZ,SearchBasisSets,SearchFunctionals,SkeletalStructure,Thermodynamics,TransitionDipolePlot,TransitionDipoles,TransitionOrbitalPlot,TransitionOrbitals,Variational2RDM,VibrationalModeAnimation,VibrationalModes,Video
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
bond_distances ≔ mapx → x/10+1.11, fsolveexpandChebyshevT6,x;
bond_distances≔1.01340742,1.03928932,1.08411810,1.13588190,1.18071068,1.20659258
We define a list of molecular geometries with each geometry corresponding to one of the bond distances
molecules ≔ seqC,0,0,0,O,0,0,R, R in bond_distances;
molecules≔C,0,0,0,O,0,0,1.01340742,C,0,0,0,O,0,0,1.03928932,C,0,0,0,O,0,0,1.08411810,C,0,0,0,O,0,0,1.13588190,C,0,0,0,O,0,0,1.18071068,C,0,0,0,O,0,0,1.20659258
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.
energies ≔ seqEnergymolecule,method=MP2, basis=cc-pV5Z, molecule in molecules;
energies≔−113.21146174,−113.22841767,−113.24555326,−113.25064671,−113.24594169,−113.24039923
We use polynomial interpolation to generate a polynomial in the bond distance R
pes ≔ interpbond_distances,energies,R;
pes≔−9.66039475R5+62.38556861R4−163.80401804R3+218.86658796R2−148.46546961R−72.52251752
The potential energy surface (curve) can be plotted
plotpesR, R=1.025..1.2, axes=boxed, labels='R','E', color=red, thickness=3;
Finally, we differential the potential energy curve with respect to R and set the derivative to zero.
eq ≔ diffpes, R = 0;
eq≔−48.30197374R4+249.54227445R3−491.41205412R2+437.73317592R−148.46546961=0
Solving the resulting equation yields the equilibrium bond length
R_eq ≔ fsolveeq, R=1.0..1.2;
R_eq≔1.13054421
(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
molecule ≔ C,0,0,0,O,0,0,1.11014350;
molecule≔C,0,0,0,O,0,0,1.11014350
Using the Dipole command, we can compute the dipole from the MP2 method in the cc-pVDZ basis set
Dipolemolecule, method=MP2, basis=cc-pVDZ;
X0.Y0.Z−0.01274002
For comparison we can also compute the dipole from the Hartree-Fock method
Dipolemolecule, method=HartreeFock, basis=cc-pVDZ;
X0.Y0.Z−0.14581806
(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
bond_distances ≔ mapx → x/1.7+1.55, fsolveexpandChebyshevT6,x;
bond_distances≔0.98180834,1.13405483,1.39775350,1.70224650,1.96594517,2.11819166
molecules≔C,0,0,0,O,0,0,0.98180834,C,0,0,0,O,0,0,1.13405483,C,0,0,0,O,0,0,1.39775350,C,0,0,0,O,0,0,1.70224650,C,0,0,0,O,0,0,1.96594517,C,0,0,0,O,0,0,2.11819166
The energies for each geometry may be then readily computed with the Energy command in the Quantum Chemistry package.
energies_mp2 ≔ seqEnergymolecule,method=MP2, basis=cc-pVDZ, molecule in molecules;
energies_mp2≔−112.95544897,−113.04050587,−112.96547532,−112.80294666,−112.60289471,−112.54447209
pes_mp2≔ interpbond_distances,energies_mp2,R;
pes_mp2≔−3.48933102R5+27.24786757R4−84.47258157R3+130.33469869R2−99.84887326R−82.74826166
Similarly, for comparison we can compute the energies with the Hartree-Fock method
energies_hf≔ seqEnergymolecule,method=HartreeFock,basis=cc-pVDZ, molecule in molecules;
energies_hf≔−112.68907086,−112.74866568,−112.62428645,−112.43823734,−112.34119827,−112.31450377
pes_hf ≔ interpbond_distances,energies_hf,R;
pes_hf≔−0.86844042R5+7.99580630R4−29.34827331R3+53.21006552R2−46.83555798R−96.85908538
The potential energy curves from the Hartree-Fock method (red) and the MP2 method (blue) can be plotted together
p_hf ≔ plotpes_hf, R=1.0..2.1, axes=boxed, labels='R','E', color=red, thickness=3: p_mp2 ≔ plotpes_mp2, R=1.0..2.1, axes=boxed, labels='R','E', color=blue, thickness=3:plots:-displayp_hf,p_mp2;
(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?
C. Møller and M. S. Plesset, Phys. Rev. 46, 618–622 (1934). "Note on an Approximation Treatment for Many-Electron Systems"
A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover Books, New York, 1996).
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