Møller-Plesset (MP) Theory, or Many-Body Perturbation Theory (MBPT) is an economical way to partially correct for the lack of dynamic electron correlation in the Hartree-Fock Theory. It improves the description of molecular energies and properties significantly in situations where the single-reference Hartree-Fock wave function already describes the system reasonably well. The second-order energy correction (MP2) is only slightly more time-consuming that evaluation of the HF energy for small and medium molecules and the MP2 approach can be meaningfully applied to systems consisting of tens of heavy atoms. The third-order correction is not widely used because it is costlier than MP2 and its advantages over MP2 in calculations with modest basis sets are not clear. The fourth-order correction (MP4(SDTQ) is applicable to smaller systems but may be required in cases when crowding of electron pairs occurs.
Correlation energies depend significantly on the basis set. In general, large correlation consistent basis sets provide best results while use of smaller basis, such as 6-31G(d) with electron correlation methods may give erroneous results. The convergence of second order correlation energies in correlation-consistent basis is well established: the two energy components (singlets and triplets) converge as X-3 and X-5, respectively. A fairly accurate estimate for the basis set limit of the second-order correlation energy can be obtained by separate extrapolation of singlet and triplet contributions from the cc-pVTZ and cc-pVQZ data according to formulas (EX = Einf + A * (X+C)-B where B = 3 for singlets and B = 5 for triplets. The additive constant depends on the ratio of heavy atoms to hydrogens. Taking C = 1 for molecules consisting only of heavy atoms and taking C = 0.5 for hydrogen-containing molecules yields reliable extrapolated second order contributions.
Some of the advantages of the MP theory include the size-consistency, interpretability in terms of correlations between specific number of electrons, and the ability to describe van der Waals attractive forces. The MP2 method shows formally a N5 scaling with the system size, but in practice better scaling can be achieved with clever algorithm design. For example, the local MP2 method that ignores correlation effects between distant electrons scales almost linearly with the size of the molecular system and is applicable for larger molecules, such as paclitaxel (Taxol, on the right) in triple-zeta basis. The MP2 level gives good structures and predicts well the IR spectra for many molecules. In particular, for many small and medium size molecules, optimization at the MP2/cc-pVTZ level gives good geometries. The disadvantages include poor convergence of the perturbation series and its non-variational nature. The latter implies that the MP energy may be lower than the true energy by an unknown amount. Also, some spectral properties, such as UV excitation energies still have very large errors when Møller-Plesset corrections are applied to single-reference wave function. However, the root cause here is the inadequate description of exited states by a single-reference wave function.
Coupled cluster (CC) theory provides a rather good description of electron correlation at the expense of longer computational time. One of the reasons that coupled cluster calculations take longer lies in the iterative nature of the CC solutions. Also, the calculation of full triples or quadruples contributions as in CCSDTQ has a very unfavorable scaling with increasing size of the system. This limits the use of such calculations to very small molecules. The CCSD(T) method, in which the triples contribution is calculated approximately, is almost as accurate as CCSDT but significantly cheaper. This level of theory still provides very accurate energies and geometries, allowing, for example, ab initio estimation of heats of formation.
Correlation energies due to double excitations depend significantly on the basis set. As a result, CCSD energies should be evaluated using good basis sets. In particular, CCSD(T)/cc-pVQZ level provides very accurate geometries in many cases. The convergence of CCSD correlation energy in correlation-consistent basis has not been fully established but the two contributions from singlets and triplets show different convergence rates, with the slowest-converging terms going as X-3. Coupled clusters theory is size consistent and may be well suited for description of transition state structures where the single-reference HF wave function may be less-than-optimal reference wave function.
Full Configuration Interaction (FCI) approach provides the best description of molecular electronic structure and energies but is applicable to only very, very small systems. Truncated configuration interaction calculations, such as CISD were once quite popular. However, the lack of size-consistency in truncated CI calculations hampers their use for describing chemical processes. Configuration Interaction calculations are naturally well suited for the description of excited states and are sometimes used for calculation of UV/Vis spectra.
Jaguar is the ab initio code in the Schrödinger program suite. Among other interesting features, it allows efficient Møller-Plesset calculations using the local MP2 approximation. You will investigate its speed advantage in a calculation of a small molecule with a small basis set. Your goal is to calculate the gauche-anti conformational energy difference in ethanol using the LMP2/6-31+G(d) method with Jaguar and compare the results with a similar calculation in Gaussian. First, build the anti conformer of ethanol in Maestro Builder, create an entry, and perform LMP2/6-31+G* optimization. Write down the final energy and calculate how long the calculation took. Then repeat this for the gauche conformer. Why is the gauche conformer calculation taking longer? Export the Gaussian Z-matrices for the anti and gauche conformer from Maestro and edit these files to perform geometry optimization with Gaussian. The input for the MP calculations is described in the Gaussian Online Manual. Use the Opt keyword without the CalcAll option here because the Newton-Rhapson method was not used for optimization with Jaguar. Submit the two Gaussian jobs to the queue. Compare the gauche-anti energy gap from the Jaguar's LMP2/6-31+G(d) calculation with the Gaussian's MP2/6-31+G(d) result. Compare the computational times. Is Jaguar faster for this problem?
Carbon monoxide, despite its simplicity, is quite challenging system due to the triple-bonded nature of the molecule. The experimental C-O equilibrium distance is 1.1283 angstroms and the dipole moment is -0.112 D (oxygen being positive). Perform a geometry minimizations at the MP2/cc-pVTZ, MP4(SDQ)/cc-pVTZ, and CCSD/cc-pVTZ levels with Gaussian to see how well each of these methods performs in the predicting the structure and the dipole moment of CO. The input for the CCSD calculations is described in the Gaussian Online Manual. To calculate dipole moment based on the MP2, MP4, or CCSD density use the keyword Density in the input. Try to perform CCSD(T)/cc-pVTZ optimization with Gaussian. Does it work?
Efficient CCSD(T) optimizations are implemented in the program Dalton. This free-for-academic-use program is geared toward calculation of a wide range of molecular properties, such as polarizabilities, UV/Vis, circular dichroism, NMR and EPR spectra. However, the structure of the Dalton input is quite different from the Gaussian input. We are learning the Dalton input because a growing number of free quantum chemistry programs use input very similar to the Dalton input.
The input geometry of molecules can be specified in two ways: Z-matrix or XYZ positions. Gaussian and Dalton accept both formats but we will illustrate the XYZ format below. For example, a CCSD/cc-pVDZ optimization of a nitrogen molecule can be specified as:
BASIS cc-pVDZ Nitrogen: CCSD/cc-pVDZ calculation DALTON Opt: XYZ input in Bohr (atomic) units: 1 Bohr = 0.5292 Ang Atomtypes=1 Charge=7.0 Atoms=2 N 0.0000000000 0.0000000000 0.0000000000 N 0.0000000000 2.0500000000 0.0000000000 **DALTON INPUT .DIRECT .OPTIMIZE **INTEGRAL .DIPLEN .DEROVL .DERHAM **WAVE FUNCTIONS .CC *CC INPUT .CCSD .FREEZE 2 0 **END OF DALTON INPUT
The input above consists of two parts: the molecule specification and directives for the calculation. The directives are grouped according to program modules. For example, the .OPTIMIZE directive tells the main Dalton program that a geometry optimization is required, and the .FREEZE directive tells the coupled cluster module that electrons on the two lowest energy orbitals (atomic core orbitals) should be excluded from the correlation treatment. A short Dalton calculation can be submitted outside the queue by dalton inputfile > log &.
Compare the result of the ethanol gauche-anti energy gap from your tutorial with the accurate experimental literature value. Perform a quantum chemical calculation that reproduces the experimental gas phase value within 0.06 kcal/mol.
Optimize the geometry of a water molecule at the MP2, MP3, MP4 levels using the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis. Make sure to use appropriate molecular symmetry to speed up calculations. The analytical Newton-Rhapson method (Opt=CalcAll) in Gaussian is implemented only at the MP2 level and you should use the default optimizer (Opt) for MP3 and MP4. Is the convergence of the H-O distance and the H-O-H angle systematic with the increasing basis set? Compare your results with the accurate experimental geometry of water in the gas phase and discuss which is the best method for predicting the geometry of water.
Determine the structure of carbon monoxide at the CCSD(T) level with cc-pVDZ, cc-pVTZ, and cc-pVQZ basis and compare results to the experimental data. Prepare a graph illustrating how the carbon-oxygen bond length varies with MP2, MP4(SDQ), CCSD, and CCSD(T) methods using the cc-pVTZ basis. Prepare a graph illustrating how the the carbon-oxygen bond length varies in CCSD(T) calculations with cc-pVDZ, cc-pVTZ, and cc-pVQZ basis.
You may use any computational chemistry program for this task. However, not all programs permit all calculations. If you prefer using Windows, then PC Gamess is a good choice for MP2 optimizations; however it presently does not support MP4 or CCSD optimizations