Self Consistent Field Theory

The Potential Energy Surface and the Hamiltonian

In the Born-Oppenheimer approximation it is assumed that the motion of the nuclei is so slow that the electrons instantaneously follow them. This is because mass(proton)$\sim$2000$\times$mass(electron). Therefore the electronic wavefunction obeys the Schrodinger equation with the nuclei at rest. This equation is
 

$$H({\bf R})\Psi=W({\bf R})\Psi$$ (79)

 where

$$H=-\frac{1}{2}\sum_i^n\nabla_i^2-\sum_{iA}^{nN}\frac{Z_A}{\vert{\...... r}_i-{\bf r}_j\vert}+\sum_{A>B}^N\frac{Z_AZ_B}{\vert{\bf R}_A-{\bf R}_B\vert}$$ (80)

 Atomic units have been used ( $e=m_e=a_o=\hbar=1$). n is the number of electrons, N is the number of nuclei A and Z_A the nuclear charges. We see that the positions ${\bf R_A}$ enter the hamiltonian. Thus the energy, $W({\bfR})$ depends upon the position of the nuclei. $W({\bfR})$ is called The Potential Energy Surface. The quantum chemist (computational chemist) attempts to calculate (sections of, pointwise) potential energy surfaces. No parameters are input into the code. The Schrodinger equation is impossibly difficult to solve, and approximations are essential.

The Electron Density

The electron density $\rho({\bf r})$ is the probability of finding an electron at ${\bf r}$. From the laws of quantum mechanics it is expressed in terms of the the square of the orbitals which contain the electrons. Thus

$$\rho({\bf r})=2\sum_i^n\phi_i^2({\bf r})$$ (81)

 where we have assumed we have two electrons in each of n orbitals, i.e. we have a 2n electron molecule. Note that by definition

$$\int\rho({\bf r})d{\bf r}=2n$$ (82)

 From studying H₂, we understand that the density $\rho({\bf r})$ is `cusped' at the nuclei and furthermore the shape of the cusp will depend upon the particular nucleus, the bigger the nuclear charge Z_A the more pronounced the cusp. In mathematical terms therefore a knowledge of the density tells you (i) where the nuclei are, (ii) what they are and (iii) how many electrons are in the molecule, from (63). It follows that in principle the density contains all information, because once ${\bf R_A}$ and Z_A are known, the hamiltonian (61) is known, and so the wavefunction is known, and so all is known.

The Energy Expression

Once the orbitals have been determined (and therefore the density) it is possible to determine the energy $W({\bfR})$ of the molecule. It is given by

$$2\sum_i\int d{\bf r}\phi_i({\bf r})(-\frac{1}{2}\nabla^2\phi_i({\......\int d{\bf r^{'}}\frac{\rho({\bf r})\rho({\bf r^{'}})}{\vert{\bf r-r^{'}}\vert}$$ W =

$$-\frac{3}{4}C\int d{\bf r}\rho^{\frac{4}{3}}({\bf r})+\sum_{A>B}\frac{Z_AZ_B}{\vert{\bf R_A-R_B}\vert}$$ (87)

 
These terms may be identified as (i) the kinetic energy of the electrons, (ii) the attractive energy between nuclei and electrons, (iii) the coulomb repulsion energy between electrons, (iv) the difficult self-interaction energy and `electron-avoidance' energy, (v) the nuclear-nuclear repulsion energy. It is important to note that the Energy is NOT the sum of the orbital energies, as may be seen from eqns(86) and (87). The sum of the orbital energies counts twice the coulomb repulsion energy.

Basis sets and the Self Consistent Equations

The orbitals are expressed as linear combinations of basis functions. Basis functions are the logical extension of atomic orbitals to a more accurate expansion set. Today basis functions are gaussian functions $x^py^qz^s\exp(-ar^2)$ situated on different atoms. Thus we write

$$\phi_i=\sum_{ki}^Mc_{ki}\eta_k$$ (88)

 M is the number of basis functions and $\eta_k$ are the basis functions. Small (STO-3G), medium (6-31G^*) and large (TZ2P) basis sets are in common use, giving results increasingly close to the accurate result.

To find the orbitals, we solve

$$\sum_j^n\langle\eta_k\vert F-\epsilon_i\vert\eta_j\rangle c_{ji}=0$$ (89)

 which is a set of secular equations as we have met before. This time we have a definition for F_kj, it is

$$F_{kj}=\langle\eta_k\vert F\vert\eta_j\rangle \langle\eta_k\vert-\frac{1}{2}\nabla^2-\sum_A\frac{Z_A}{\vert{\bf R_A-r}\vert}-C\rho^{\frac{1}{3}}({\bf r})\vert\eta_j\rangle$$ =

$$\int d{\bf r}\ \eta_k({\bf r})\eta_j({\bf r})\int d{\bf r^{'}}\frac{\rho({\bf r^{'}})}{\vert{\bf r-r^{'}}\vert}$$ + (90)

 The important point to observe is that there is an expression for the F matrix, it can be evaluated, and it involves the orbitals which we are trying to find. Thus the equations must be solved by an iterative procedure. A typical output is attached, showing the convergence of the energy with the iterations, the converged orbital energies and the molecular orbital coefficients.
In this formaldehyde example note that:
(i) orbital 1 is 1s_O and is very low in energy
(ii) orbital 2 is 1s_C and is again low in energy
(iii) orbital 3 is 2s_O. It does not participate in bonding.
(iv) orbital 4 is bonding CH₂,involving 2s_C, 2p_zC and 1s_A+1s_B, as well as some 2s_O to preserve orthogonality
(v) orbital 5 is bonding CH₂, b₂ symmetry, constructed from 2p_yC and 1s_A-1s_B.
(vi) orbital 6 is $\sigma$ bonding between 2p_zO and 2p_zC
(vii) orbital 7 is $\pi$ bonding between 2p_xO and 2p_xC
(viii) orbital 8 is nonbonding 2p_yO lonepair.
(ix) observe that the sum of the orbital energies is not equal to the energy of the molecule. There is double counting in the electron-electron interactions in the former
(x) The molecule is positively charged at the CH₂ end, and negatively charged at O, leading to the dipole moment of 1.54D
(xi) the orbital energies are reasonable approximations to the ionisation energies

Density Functional Theory

The above are the working equations for the molecular orbitals and energy of a molecule, using the theory which is today called `Density Functional Theory' (DFT). The expression for the energy involves the density as anticipated. The success of the theory depends upon how well the difficult term $v_{xc}({\bf r})$ can be represented in terms of the density, the above only gives the most simple approximation to it. It is to be stressed that the hamiltonian F is the same for all molecules and contains no adjustable parameters. It is also important to realise that there is no wavefunction in DFT, but only the density. Nasty things such as determinantal wavefunctions and exchange integrals do not exist in DFT. In this sense DFT is an ab initio theory. 90% of ab initio molecular calculations use DFT today, the remainder use much more difficult theories which attempt to approximate the electronic wavefunction, which we have seen is a much more complicated object (a (linear combination of) determinant(s)). As a measure of the reliability of DFT predictions, typical dissociation energies ( W(AB)-W(A)-W(B)) are calculated to an accuracy of 10 kJ mol^-1, molecular bondlengths to 0.01Å, and vibrational frequencies to 2%. DFT is an example of the Self Consistent Field (SCF) method, where the orbital equations have to be iterated until self-consistency is achieved.

There are now many standard quantum chemistry packages which routinely perform DFT and more sophisticated calculations. Quantum chemistry is now an accepted predictive tool of chemistry, used by organic, inorganic and physical chemists. More in the T2 option!

Finally as an example of DFT, let us reexamine the proof of Hund's rules in the simple case of the 1s2s excited state of Helium. All we concern ourselves with is the exchange energy which according to DFT is given by

$$E_X=-C\int(\rho_{\alpha}^{\frac{4}{3}}({\bf r})+\rho_{\beta}^{\frac{4}{3}}({\bf r}))d{\bf r}$$ (91)

 where $\rho_{\alpha},\rho_{\beta}$ are the alpha and beta spin densities. In the singlet case $\rho_{\alpha}=1s^2,\rho_{\beta}=2s^2$ and in the triplet case $\rho_{\alpha}=1s^2+2s^2,\rho_{\beta}=0$. Thus the exchange energies in the two cases are

$$E_X({\rm singlet})=-C\int((1s^2)^{\frac{4}{3}}+(2s^2)^{\frac{4}{3}})d{\bf r}$$ (92)

$$E_X({\rm triplet)}=-C\int(1s^2+2s^2)^{\frac{4}{3}}d{\bf r}$$ (93)

 It follows that the triplet is lower in energy than the singlet because it can easily be proved that $(x+y)^{\frac{4}{3}}>x^{\frac{4}{3}}+y^{\frac{4}{3}}$.

1998-09-23