Introduction to computational chemistry Exercise 1: Structure and dipole moment of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the dipole moment of a small molecule (or molecules) are calculated using various ab initio methods. This exercise has three different purposes: to familiarize you with various ab initio methods and to let you understand their accuracy and their capabilities with respect to experimental data. to familiarize you with various ab initio basis set acronyms, to give you some hints on the methods which are used to obtain them, and to give you an idea of the computational efforts to obtain accurate results. to practise with tools and programs commonly used in computational chemistry research. 2 Ab initio methods The term ab initio means from first principles. It does not mean we are solving the Schrödinger equation exactly. It means that we are selecting a method that, in principle, can lead to a reasonable approximation to the solution of the Schrödinger equation, and then selecting a basis set that will implement that method in a reasonable way. By reasonable, we mean that the results are adequate for the application at hand. A method and basis set that is adequate for one application may be inadequate for another. We also have to take into account the cost of doing calculations and the total amount of time required. A wide range of methods have been employed, but in this exercise we will restrict ourselves to the density funtional method and some commonly used methods that use molecular orbital theory (i.e. Hartree Fock). The methods used in this exercise are the following: RHF MP2
QCISD CCSD(T) B3LYP The RHF method is the restricted Hartree Fock method, where the electrons are paired, so they can be treated two at a time. The next three methods try to improve the Hartree Fock electronic wavefunction: the MP2 (Møller Plesset) method, employs perturbation theory at the second order. The QCISD (Quadratic Configurations Interaction of Single and Double excitations) calculates the electronic wavefunction as a linear combination of Hartree Fock determinants in which all the single and double excitations are included. The CCSD(T) (Coupled-Cluster Single, Double, and perturbative Triple excitations) is based on an exponential approach. B3LYP is a density functional method; the electronic energy is expressed using the electronic density, not the wave function. 3 Basis functions Historically, quantum calculations for molecules were performed as LCAO MO, i.e. Linear Combination of Atomic Orbitals - Molecular Orbitals: ψ i = n c ij φ j (1) j=1 where ψ i is the i-th molecular orbital, c ij are the coefficients of linear combination, φ j is the j-th atomic orbital, and n is the number of atomic orbitals. Atomic Orbitals (AO) are solutions of the Hartree Fock equations for the atom, i.e. a wave functions for a single electron in the atom. More recently, the term atomic orbital has been replaced by basis function or contraction, when appropriate. Slater-Type-Orbitals (STOs) are similar to the AOs of the hydrogen atom. They are described by a function that depends on spherical coordinates: φ i (ζ, n, l; r, θ, φ) = Nr n 1 e ζr Y lm (θ, φ) (2) where N is the normalization constant and ζ is the exponential factor. The variables r, θ, and φ are spherical coordinates, and Y lm is the angular momentum part (function describing shape ). The integers n, l, and m are quantum numbers: principal, angular momentum, and magnetic, respectively. Unfortunately, STOs are not suitable for fast calculations of necessary two-electron integrals. That is why the Gaussian-Type-Orbitals (GTOs) were introduced. They can be used to approximate the shape of the STO function by summing up a number of GTOs with different exponents and coefficients. You can use several GTOs (4 5) to 2
represent an STO, and still calculate integrals faster than if the original STOs were used. The GTO is expressed as: φ GT O (α, l, m, n; x, y, z) = Ne αr2 x l y m z n (3) where N is the normalization constant and α is the exponential factor. The variables x, y, and z are cartesian coordinates. The quantities n, l, and m are NOT QUANTUM NUMBERS but simply integral exponents at cartesian coordinates. The GTOs are not really orbitals, they are simpler functions. In recent literature they have been called gaussian primitives. How are the GTOs obtained? The following example shows how the variational principle can be employed to find the best possible basis function expressed by single gaussian function to the lowest symmetric level for hydrogen atom. Example 1. The exact eigenfunction corresponding to the 1s hydrogen atomic orbital (for which the corresponding energy is 0.5 Hartree), is given by the function: φ 1s = 1 π e r (4) It was mentioned earlier that in ab initio calculations, it is customary to approximate the monoelectronic functions as linear combinations of gaussians centred on the atoms, that is as linear combinations of functions of the kind: φ GF = Ne αr2 (5) where N is the normalization constant and α is unknown parameter. In this example we are limited to use only one gaussian in the trial function in order to calculate the lowest energy of hydrogen, which can be obtained by using the radial part of the Schrödinger equation when the angular momentum is zero: where E = φ GF 1 2 2 1 φ GF r (6) 2 = 1 ( d r 2 d ) r 2 dr dr (7) The variation principle states that the energy calculated by your guessed wavefunction will be greater than (or equal to) the actual ground state energy. Thus, the problem is to find a value for the unknown parameter α that minimizes the energy. It turns out that the lowest energy using the trial function of Eq.(5) is E = 4 Hartree = 0.4244 Hartree, whereas 3π the exact value is E = 0.5 Hartree. Another more general approach to obtain basis functions is to numerically find such a linear combination of gaussian functions that maximizes the overlap between it and the Slater functions. In other words, the Slater functions are fitted 3
using linear combinations of gaussian functions. These functions are called STOnG basis functions where n refers to the number of gaussians used to express the Slater-Type-Orbital. For molecular calculations, it is obvious that the best results could be obtained if all coefficients in the gaussian expansion were allowed to vary during the calculation. In practice however, the gaussian primitives have to be contracted. The term contraction means a linear combination of gaussian primitives to be used as a basis function. Such a basis function will have its coefficients and exponents fixed. Minimal basis sets are not enough to describe the shapes of the molecular orbitals correctly. Thus, several basis functions (or contractions) are required for a more accurate treatment of electronic states in molecules. Basis sets that possess more than one contraction to describe each electron are called extended basis sets. There are several types of extended basis sets: Double-Zeta, Triple-Zeta,... Multiple contractions are used to represent a single Slater orbital. Split-Valence Multiple contractions are used for only the valence (outer) orbitals. Polarized Sets The basic picture of atomic orbitals existing only as s, p, d, f etc. is modified by mixing the different types. This treatment takes into account the polarization effect which distorts the shape of the atomic orbitals when atoms are brought together. Diffuse Sets These basis sets utilize very small exponents to clarify the properties of the tail of the wave function. When an atom is in an anion, in an excited state or when long range effects are important for some other reason, the tail of the wave function becomes important. 4 Exercise In this exercise, the program Gaussian03 will be used to calculate the optimised structure and the dipole moment of a molecule (or molecules). First, prepare the input file jobname.com. An example of the Gaussian input is as follows: 4
#RHF/STO-3G Opt=z-matrix Water equilibrium geometry 0 1 O H1 O r1 H2 O r2 H1 a1 r1 0.95 r2 0.95 a1 105.0 In the first line, after the symbol #, the ab initio method and the basis set are specified. For the various methods, it will be sufficient to change the name of the method or the basis set (that is, MP2/AUG-cc-pVTZ for example). In the second line, the job type is specified. In this exercise, the geometry of a molecule is optimised. This is implemented with the Opt keyword in the input. The option z-matrix requests that the optimisation be performed in internal coordinates. The fourth line is reserved for the title which will be written in the output file. In the sixth line, the charge and spin multiplicity of the molecule are specified, respectively. In the following three lines, the z-matrix, i.e the geometry in internal coordinates, is specified in Å for bond lengths and degrees for bond angles and dihedral angles. After the empty line, the initial values for the coordinates are given. In the optimisation, the initial values are varied by Gaussian until the energy minimum is found. The exercise will be carried out using the methods B3LYP, RHF, MP2, QCISD, and CCSD(T) and at least the following basis sets: STO-3G: Minimal basis set. Each Slater-Type-Orbital expressed by 3 gaussians. 6-31G(d) (or 6-31G*): Pople s (Split Valence) basis set which includes 6 gaussian primitives for the inner shells and uses the 31 contraction scheme for the valence electrons. In this contraction scheme each orbital is described by two basis functions, the first of which includes three gaussian primitives and the second of which contains one. Adding a single polarization function to 6-31G (i.e. 6-31G(d)) will result in one d function for first and second row atoms and one f function for first transition row atoms, since d functions are already present for the valence electrons in the latter. AUG-cc-pVTZ: Dunning s correlation consistent basis set. Includes polarization functions by definition. The term TZ (triple zeta) means that three 5
contractions are used to represent a single Slater orbital. This basis set is augmented with diffuse functions by including the AUG- prefix to the basis set keyword. However, the elements He, Mg, Li, Be, and Na do not have diffuse functions defined within this basis set. In Corona, the Gaussian job is submitted by the command: subg03 00:20:00 jobname The dipole moment and the optimised geometry will be written in the end of the jobname.log file. For example, you can organise your results (for each coordinate and dipole moment) in a table of the following type: STO-3G 6-31G(d) AUG-cc-pVTZ RHF MP2 QCISD CCSD(T) * * is too resource heavy for the purpose of this exercise. Experimental structures and dipole moments of some molecules: CO dipole moment: 0.112 D r=1.11 Å H 2 O dipole moment: 1.85 D r=0.9584 Å a=104.45 degrees H 2 O 2 dipole moment: 2.26 D r(o-h)=0.96±0.01 Å r(o-o)=1.47 Å a(o-o-h)=97.0±2 degrees a(dihedral)=111.7±0.1 degrees NH 3 dipole moment: 1.42 D r(n-h)=1.017 a(h-n-h)=107.5 degrees 6