Basis functions in an atomic basis set can be characterized by the principal quantum number (n) and the angular quantum number (l). The principal quantum number characterizes the size (radial extent) of the orbital; the angular quantum number describes its shape. For example, the spherical basis function that describes the 1s atomic orbital in the hydrogen atom has the principal quantum number n = 1 and the orbital quantum number l = 0. A balanced description of electron distribution in atoms and molecules is achieved by a basis sets that include some functions with high principal quantum number and some functions with high angular quantum number. Such basis functions are especially important to describe the dynamic electron correlation, which is ignored in the Hartree-Fock theory. For example, some advanced correlated calculations of the Ne atom have employed bases providing a 25s orbital as well as some k-orbitals (l=7).
Ideally, a basis that has many functions with large principal quantum number as well as with large angular quantum number should be employed. In practice, such complete bases are computationally too taxing and may suffer from linear dependencies. One practical solution was suggested by Thom Dunning in 1989: let's employ a family of basis sets, in which each member systematically improves on the previous member. The correlation consistent basis sets of Dunning are constructed such that each basis in the series supplements the previous basis with a complete shell of functions that go with the current principal quantum number. For example, the carbon basis set with the maximum principal quantum number lmax = 4 adds a single s, p, d, f, and g function to a basis that already has four s, three p, two d, and one f function. Traditionally, bases in the cc-pVXZ series are characterized by the cardinal number X, which is related to the maximum angular momentum function present in the basis set For example, the largest angular momentum function in the cc-pV5Z basis (X=5) for carbon is h, thus lmax=5 and X = lmax . However, the largest angular momentum function in the cc-pV5Z basis for hydrogen and helium is g, thus lmax=4 and now X = lmax + 1.
Consider a set of calculations performed on the same molecule but using a series of correlation consistent basis sets. It turns out that the Hartree-Fock energy converges exponentially in such a series and the limiting value can be obtained by extrapolation. Because the exponential decay (EX = Einf + A * exp(-B*X) is described by three parameters (Einf, A, and B), three energy values (say, with cc-pVDZ, cc-pVTZ, and cc-pVQZ bases) are needed to extrapolate Einf. In practice, good results are obtained by using cc-pVTZ, cc-pVQZ, and cc-pV5Z data.
Calculate the Hartree-Fock energy at the basis set limit for the He atom based using correlation consistent basis sets up to X = 6. Provide a graph illustrating the convergence of the energy with respect to the cardinal number. Compare your result with an accurate literature value; provide the literature reference.
Calculate the rotational barrier in ethane at the basis set limit using the Hartree-Fock method. You may optimize the structures of the staggered minimum and the eclipsed saddle point at the HF/6-31+G(d,p) level. If you employ the symmetry correctly, the structure of the saddle point can be obtained by usual Newton-Rhapson minimization, otherwise Opt=(CalcAll,TS) can be used to reach the saddle point. Analyze the output of the saddle point optimization job to verify that you indeed have one negative frequency and one negative eigenvalue in the Hessian. Use cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z basis sets for single point energy calculations. Calculate the Hartree-Fock limit for the minimum and the saddle point and obtain the rotational barrier by subtracting the energy of the minimum from the energy of the saddle point. Provide graphs illustrating the convergence of the absolute energies with respect to the cardinal number. Also provide a graph illustrating the convergence of the rotational barrier of the water dimer with respect to the cardinal number. Compare your result with an experimental rotational barrier of ethane and comment on the accuracy of the HF calculations for estimating methyl group rotational barriers.
Calculate the stability of the water dimer at the basis set limit using the Hartree-Fock method. You may optimize the structures of isolated H2O and the dimer in the the HF/6-311++G(d,p) basis. If you use the Newton-Rhapson algorithm, start from a good initial structure (feel free to consult literature on how water dimer looks), then verify that your Hessian has only positive eigenvalues. Use cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-PV5Z basis sets for single point energy calculations. Calculate the Hartree-Fock limit for the isolated molecules and the dimer. Provide graphs illustrating the convergence of the absolute energies with respect to the cardinal number. Also provide a graph illustrating the convergence of the stability of the water dimer with respect to the cardinal number. Compare your result with an experimental stability of the water dimer and comment on the accuracy of the HF calculations for estimating strengths of intermolecular complexes.
You may use a computational chemistry program of your choice. If you prefer using Windows, then PC Gamess is a good choice (however, you still want to build your molecules in MOLDEN).