Property Basis Sets

A completely different type of property is for example spin-spin coupling constants, which contain information about the interactions of electronic and nuclear spins. One of the operators is a 5 function (Fermi contact, eq. (10.73)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp  

Yamaguchi, Y. Osamura, J. D. Goddard, H. F. Schaefer 111, 4 New Dimension to Quantum Chemistry, Oxford University Press, 1994 C. E. Dykstra, J. D. Augspurger, B. Kirtman, D. J. Malik, Rev. Comp. Chem., 1 (1990), 83 D. B. Chesnut, Rev. Comp. Chem., 8 (1996), 245 R. McWeeny, Methods of Molecular Quantum Mechanics, Academic Press, 1992 J. Olsen, P. J0rgensen, Modern Electronic Structure Theory, Part II, D. Yarkony, Ed., World Scientific, 1995, pp. 857-990. 

Vahtras, O. Loboda, B. Minaev, H. Agren, K. Ruud, Chem. Phys., 279 (2002), 133. 

A ((P Q)) propagator is called a linear response function, since it measures the response of P to a perturbation Q. The ((r r)) propagator thus determines the polarizability, which is a second-order property. The concept may be generalized to higher orders, i.e. the quadratic response function, given as a ((P Q,R)) propagator. 

Helgaker, M. Jaszunski, K. Ruud, and A. Gorska, Theor. Chem. Acta, 99 (1998), 175. 

In this chapter we will illustrate some of the methods described in the previous sections. It is of course impossible to cover all types of bonding and geometries, but for high-lightirig the features we will look at the H2O molecule. This is small enough that we can employ the full spectrum of methods and basis sets. 

More recently Equation Of Motion (EOM) methods have been used in connection with other types of wave functions, most notably coupled cluster.Such EOM methods are closely related to propagator methods, and give working equations which are similar to those encountered in propagator theory. 

The basis set requirements for obtaining a certain accuracy of a given molecular property are usually different from those required for a corresponding accuracy in 

---

Scott, A. R Radom, L. Harmonic vibrational frequencies an evaluation of Hartree-Fock, Moller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors, J. Phys. Chem. 1996,100, 16502-16513. Halls, M. D. VeUcovski, J. Schlegel, H. B. Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set, Theor. Chem. Acc. 2001,105, 413-421. 

In order to get accurate properties, large and carefully chosen basis sets are needed. The hyperpolarizabilities are sensitive to the tail, or outer regions, of the wavefunction, and so higher order diffuse polarization functions are needed to converge the properties. Basis sets that give good results for lu, may be poor for a and useless for In general correlated calculations demand more of a basis set than uncorrelated ones. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

Regarding mechanical properties of polymers, the efficiency of the Car-Parrinello approach has enabled us to evaluate the ultimate Young s modulus of orthorhombic polyethylene, and demonstrate basis set convergence for that property. 

The molecular electronic polarizability is one of the most important descriptors used in QSPR models. Paradoxically, although it is an electronic property, it is often easier to calculate the polarizability by an additive method (see Section 7.1) than quantum mechanically. Ah-initio and DFT methods need very large basis sets before they give accurate polarizabilities. Accurate molecular polarizabilities are available from semi-empirical MO calculations very easily using a modified version of a simple variational technique proposed by Rivail and co-workers [41]. The molecular electronic polarizability correlates quite strongly with the molecular volume, although there are many cases where both descriptors are useful in QSPR models. 

Any set of one-eleelrori ftinctions can be a basis set in the IjCAO approximation. However, a well-defined basis set will predict electron ic properties using fewer leriii s th an a poorly-defiri ed basis set. So, choosin g a proper basis set in ah inilio calculation s is critical to the rcliabililv and accuracy of the calculated results. 

The success of simple theoretical models m determining the properties of stable molecules may not carry over into reaction pathways. Therefore, ah initio calcii lation s with larger basis sets ni ay be more successful in locatin g transition structures th an semi-empir-ical methods, or even methods using minimal or small basis sets. 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

The energy obtained from a calculation using ECP basis sets is termed valence energy. Also, the virial theorem no longer applies to the calculation. Some molecular properties may no longer be computed accurately if they are dependent on the electron density near the nucleus. 

---