Lecture 2 Molecular System For a molecular system, Ψ is a function of the positions of the electrons and the nuclei in the molecule, which are denoted by irr, and IRr, respectively. We treat the electrons individually, and but each nucleus as an aggregate, rather than individual component nucleons (protons and neutrons) individually. The reason is because we are interested in the electronics states of the molecules. In order to determine Ψ, we need to start with the Hamiltonian of the molecular system, which consists of kinetic (T) and potential energy (V) terms, eneennenVVVTTH ++++=ˆˆˆ (2.1) where subscripts, n and e, are used to denote terms for nuclei and electrons. We are ignoring some small terms that depend on the spin of the electrons and nuclei. The first two terms are kinetic energy operators of the electrons and nuclei, given by 21212ˆIIInMT ∇−=∑=h (2.2) and ∑=∇−=1222ˆiiemTh. (2.3) Vnn is nuclear-nuclear repulsion, given by ∑∑<=IIJIJJInnReZZV2041πε. (2.4) Vee is electron-electron repulsion, given by ∑∑<=iijijeereV2041πε. (2.5)Ven is electron-nuclear attraction, given by ∑∑−=iIiIIenreZV2041πε. (2.6) Atomic Units The fundamental quantities, such as energy and length are usually expressed in atomic units to simplify the equations. The atomic unit of length is the Bohr radius: nmmea 052917725.0220==h. Coordinates can be transformed to bohrs by dividing them by a0. Energies are measured in hatrees, defined as 02 1aehatree = . Masses are in terms of electron mass. In terms of atomic units, the Hamiltonian of H atom is rH1212−∇−= (2.7). How to express Eqs. 2.2-2.6 in terms of atomic units? The Born-Oppenheimer approximation The Born-Oppenheimer approximation is the first of several approximations often used to simplify the task of solving the Schrodinger equation. It separates the electronic motions from the nuclear motions. This is a reasonable approximation because the mass of a typical nucleus is thousands of times greater than that of an electron. As a result, the nuclei move very slowly in comparison to the electrons, and the electrons reactessentially instantaneously to changes in nuclear positions. In other words, to electrons the nuclei look as if fixed in space at a given moment and the electronic motion can be described as occurring in a field of fixed nuclei. Using the Born-Oppenheimer approximation, we can solve the electronic motion by neglecting the kinetic energy of the nuclei. The electronic Hamiltonian is thus given by eneenneelecVVVTH +++=ˆˆ, (2.8) with which the Schrodinger equation describing the motion of electrons in the field of fixed nuclei: ).,(),(ˆRrERrHeleceffelecelecrrrrψψ= (2.9) Eeff determined from equation 2.9 is a function of the nuclear positions. Accordingly Eeff is used as the effective potential field for the nuclei Hamiltonian, )(ˆˆRETHeffnnuclr+= . (2.10) This Hamiltonian describes the vibrational, rational and translational states of the nuclei. The LCAO (tight binding) approximation to Molecular Orbitals A molecular orbital is the wavefunction of an electron under the influence of all the potentials (due to nuclei and other electrons). For a molecule consisting of many atoms, the nuclear potential alone is a complicated function and the electron potential is a lot worse (why?). Thus solving the Schrodinger equation exactly by including all the potentials is not practical. Even if this could be done, the molecular orbitals would then be obtained as a function of x, y and z (ψ(x,y,z)). We have seen that ψ(x,y,z) is complicated even for the simple H atom, it is unlikely for us to express ψ(x,y,z) in terms of analytical functions formolecules. The only possibility is numerical solutions, which means to present a table of ψ(x,y,z) in discrete points in space. If we use a grid of 100 points in each dimension, we would have 1000,000 values. This is clearly not insightful. An alternative method is to present complicated ψ(x,y,z) in terms of simpler function whose properties are well known. Fourier expansion is a technique to express complicated but periodic functions in terms of sine and cosine functions. For a one-dimensional function with period L, the Fourier expansion is )}2sin()2cos({)(0LxnbLxnaxfnnnππ+=∑∞= (2.11) , where sine- and cos-functions are referred to as the basis for the expansion. So instead of tabulating f(x) at different values, we can simply list the coefficients, an and bn of the expansion. In many cases, we may obtain a satisfactory approximation to f(x) by listing the first few coefficients, which greatly simplifies the problem. Using the same strategy, we wish to express a molecular orbital, ψ(x,y,z), as ),,(),,( zyxczyxnnnφψ∑= . (2.12) Clearly, we need to find a basis set of functions, φn, for which a good representation of the molecular orbital can be obtained by a relatively small number of terms. The most widely used basis functions are the atomic orbitals of the component atoms. In other words, the molecular orbital is represented as a Linear Combination of Atomic Orbitals, or LCAO approximation.We can see that this is a good choice for two reasons. First, when the atoms are pulled far apart, the molecular orbitals must go smoothly into the set of the atomic orbitals of the component atoms. Second, because of the proximity effect, an electron in the molecule interacts more strongly with its own nucleus, so the functional form of a molecular orbital near a nucleus must be similar to that of an atomic orbital of that atom. Application of LCAO to H2+ For a qualitative or semi-quantitative analysis, we want to include smallest possible terms in the expansion, or called minimal basis approach. Let us take a look of hydrogen ion, H2+. A minimal basis for the molecular orbital are the two hydrogen 1s atomic orbitals, ),exp(),exp(bbaarCrC−=−=φφ (2.13) where C is a normalization factor and ra and rb are the distances of the electron from the two nuclei, expressed in atomic units. The molecular orbitals of H2+ is (in this minimal basis) bbaaccφφψ+=. (2.14) The probability density of an electron in the molecular orbital is then bababbaaccccφφφφψ222222++= . (2.15) Because of the symmetry of the H2+, we expect that 22bacc = , which gives bacc ±= . So we have two molecular orbitals, given by))((),)((baaubaagucgcφφψφφψ−=+=, (2.16) where the suffixes g and u are used for symmetric and antisymmetric orbitals.