Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen

Transcription

1 Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule Vesa Hänninen

2 1 Introduction In this exercise the equilibrium structure and the electronic energy 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) 1

3 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: n ψ i = c ij φ j (1) j=1 whereψ i isthei-thmolecularorbital, c ij arethecoefficientsoflinearcombination, 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 2

4 with different exponents and coefficients. You can use several GTOs (4-5) to represent an STO, and still calculate integrals faster than if the original STOs were used. The GTO is expressed as: φ GTO (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 GTOs are not really orbitals, they are simpler functions. In recent literature they have been called gaussian primitives. In order to describe the shapes of the molecular orbitals correctly, several basis functions(or contractions) are required for accurate treatment 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. 3

5 4 Exercise In this exercise, the program Molpro will be used to calculate the optimised structure and the electronic energy of a molecule (or molecules). First, prepare the input file jobname.com. An example of the Molpro input is as follows: ***, H2, ang geometry={ h1; h2,h1,r12;} r12= basis=sto-3g hf ccsd(t) optg put,molden,nh3h2o.molden; First line describes the job. Geometry block describes the geometry of the molecule in internal coordinates. Bond lengths are given in ångstroms as indicated by ang keyword in the input. In this exercise, the geometry of a molecule is optimised. This is implemented with the optg keyword in the input. In the optimisation, the initial value is varied by Molpro until the energy minimum is found. r12 is the initial bond length. Basis set is defined after basis= keyword. Method is defined with a suitable keyword. All methods that need hf procedure (like mp2) need a line hf before the actual method. Tag put,molden,nh3h2o.molden tells Molpro to write input file for the visualization program. The exercise will be carried out using the methods B3LYP, RHF, MP2, QCISD, and CCSD(T) and for example some of 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 4

6 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-pVXZ(X=D,T): Dunning s correlation consistent basis set. Includes polarization functions by definition. The terms DZ (double zeta) and TZ (triple zeta) mean that two (DZ) and three (TZ) contractions are used to represent a single Slater orbital. These 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 Hippu, the Molpro job is submitted by the command: molpro jobname.com. At the beginning of each session Molpro package needs to be loaded by the comman module load molpro. 5

7 Experimental structures of some molecules: H 2 r= Å CO r= Å H 2 O r= Å a= degrees H 2 O 2 r(o-h)=0.967 Å r(o-o)= Å a(o-o-h)= degrees a(dihedral)= degrees NH 3 r(n-h)=1.016 a(h-n-h)=106.7 degrees 6