[For M.Sc. Chemistry 3rd Semester Students under GU, prepared by Rituraj Kalita]

Link to Potential Energy Surface (PES) and Chemical Reactions Dynamics

Link to Perturbation MO theory and the Frontier Orbital Concept

Link to Basis Sets in Hartree-Fock (or Other Ab Initio) Computations

 

1.1. The Complete Molecular Schrodinger Equation and the Complete Wavefunction

A molecule has several particles, namely its constituent nuclei and its constituent electrons, each of which moves in three dimensional space. So, the complete non-relativistic Hamiltonian for a molecule must contain kinetic energy (KE) operator terms (–ħ²/2m) for all of these constituent particles, in addition to the potential energy (PE) terms for interaction among themselves. This requirement is, however, not satisfied in the familiar H-atomic (electronic) Hamiltonian {i.e., (–ħ²/2m_e) – e²/(4pe_or_e)} that we see here and there (i.e., here the nuclear KE term (–ħ²/2m_n) is obviously missing) -- this is because here the electronic motion was only discussed with an implicit assumption about a fixed nucleus, this assumption being akin to the Born-Oppenheimer approximation discussed in next section. 

1.2. Separation of Electronic and Nuclear Motions: The Born-Oppenheimer Approximation

1.3. The Nuclear Schrodinger Equation: Translational, Rotational and Vibrational Motions

1.4. The Electronic Schrodinger Equation: MO and VB Approaches to the Electronic Wavefunction

1.5. Expressing Molecular Orbitals: The LCAO-MO Approach

2.1. Introduction to Hartree-Fock Theory:

In Hartree-Fock theory, each electron is approximated to occupy a one-electron spin-orbital (SO) [consistent with the orbital concept that this theory presumes]. Here each electron is supposed to move in the field created by the presence of the nuclei, the other electrons and also itself. So the orbitals that the electrons occupy are results of the potential field that they create while moving within these orbitals themselves, and therefore this theory is called an self-consistent field (SCF) theory.  Formerly, in the earlier Hartree theory, an electron was supposed to move in the field of the nuclei and the other electrons with each electron occupying a definite orbital, but the advent of the concept of electron indistinguishability taught us that each electron could occupy any occupied-SO, and that the particular electron we've considered can't be properly distinguished from the other ones, so that the field that it experiences is contributed even by its own presence. In Section 2.2 is detailed the said field and the associated one-electron eigenvalue equation (the Hartree-Fock equation) governing the motion of an electron (and giving the one-electron orbital u_i as its solution). 
Note: Usually an SO (here u_i) is a product of a spatial orbital f (say, a molecular orbital) and either the a or b spin function.

The whole-molecular electronic wavefunction (Y_el) is however approximated as a Slater Determinant (SD, Fig. 1) D of the n occupied spin-orbitals (SOs) u_i [i = 1,2,....,n -- where n is the total number of electrons in the molecule]. Such an wavefunction would naturally satisfy the Pauli antisymmetry principle (as interchanging any two electrons would mean interchanging two rows of that determinant, thus reversing its value). It also satisfies the electron indistinguishability criterion: note that each electron is associated with every SO within the above wavefunction expression (Fig. 1).
Notes: (i) The set of SOs are chosen to be an orthonormal one (i.e., these SOs are chosen to be mutually orthogonal ones, while each of them being a normalised one). (ii) The above SD is also denoted in an abbreviated form |u₁  u₂ ...... u_n| -- note that in this form, the normalisation constant 1/(n!)^0.5 remains implied! (iii) Here we are assuming discussion of a molecule, but Hartree-Fock study of the simpler atomic system is also possible, where the SOs get constructed from AOs instead of MOs, and the core operator potential part (–_aå A_a/r_1a, in Section 2.2) within the Hartree-Fock equation becomes simpler (i.e., just –A_a/r_1a).

 

 

2.2. The Hartree-Fock Equation:

 

The Hartree-Fock equation (also called the canonical Hartree-Fock equation?) is stated as:
            (1) u_i (1) = e_i u_i (1)     where the Fock operator is: (1) = (1) + _j=1{j (1) j (1)}

Here the core operator , the coulomb operator j and the exchange operator j are defined as: 

         (1) = – ½ – _aå A_a / r_1a

         j (1) u_i (1) = { ∫ u_j* (2) (1/ r₁₂) u_j (2) dt₂} u_i (1)

         and  j (1) u_i (1) = { ∫ u_j* (2) (1/ r₁₂) u_i (2) dt₂} u_j (1)

Notes: (i) You may note the use of atomic unit system herein, as per which the numerical values of h/2p, m_e, e² and 4pe_o are all unity (so they are not seen here). (ii) means the (d²/dx₁² + d²/dy₁² + d²/dz₁²) operator involving the 1st-electronic Cartesian coordinates (x₁,y₁,z₁), A_a means the atomic number of the a-th nucleus (equaling its number of constituent protons), whereas dt₂ indicates the space-spin infinitesimal volume element for the 2nd electron (i.e., dt₂ = dv₂ dw₂ where dv₂ = dx₂ dy₂ dz₂ and dw₂ is the 2nd-electronic spin coordinate). (iii) In the Hartree-Fock equation above, the 1st electron is used, but as the electrons are indistinguishable, this equation could have been expressed in terms of any one of the existing electrons. (iv) Similarly, the coulomb and the exchange operators are defined in terms of the 2nd electron and the 1st, but any two existing electrons could have been used to define them.

 

Significances: The Hartree-Fock equation signifies that an electron (say, electron 1) of mass m_e moves along the three dimensions within the molecule (so that its kinetic energy operator is  – ½) under the attractive potential field
–_aå A_a/r_1a due to the (several) nuclei a, and the repulsive potential field _j=1{j(1)j(1)} due to the other electrons occupying the various (j-th) occupied SOs. The core-operator potential energy part –_aå A_a/r_1a gives the potential energy of Coulombic attraction between all the nuclei and the electron 1 lying at distances r_1a from the a-th nucleus. The Coulomb operator j(1) signifies the Coulombic repulsion between the electron 1 (presently occupying the i-th SO) and the charge distribution {u_j* (2) u_j (2) dt₂} for another electron (electron 2) presently occupying the j-th SO; naturally this operator (along with the j one) gets summed over all the occupied SOs (for the molecule) within the above term _j=1{j(1)j(1)}. However, the physical significance of the exchange operator j(1) is not similarly obvious, as it has no simple classical analogue: This operator arises from the antisymmetry and electron indistinguishability requirement of the molecular electronic wavefunction, and so exchanges (i.e., switches) the spin-orbitals u_i and u_j, although otherwise operating similar to the Coulomb operator (do check their definitions above to understand these significances). 

 

The Iterative Procedure of Solution: Because of its aforesaid SCF nature, the Hartree-Fock equation has to be solved via a peculiar iterative procedure. As we find above, the Fock operator includes the j and the j operators, the definitions of which clearly requires the prior knowledge of the orbitals u_j. Thus the aim of the Hartree-Fock equation is to find the orbitals as its solution, but to construct this equation itself, we require the prior knowledge of these orbitals! To resolve this difficulty, one starts with a starting set of the orbitals, and constructs a starting Fock operator. The Hartree-Fock equation is then set up and solved to give a new set of orbitals. This then results in a new Fock operator, a new Hartree-Fock equation and a newer set of orbitals. This procedure is thus repeated again and again till one arrives at a set of orbitals practically identical with the earlier set. 

 

In practice, however, the Hartree-Fock equation is routinely solved in a similar iterative way only in the form of its linear equations-set representation, known as the Roothan equations (Section 2.7).  
Notes: To understand what is meant by an iterative procedure in mathematics (in particular, in numerical analysis) consider the simple iterative procedure for solving the equation (in radian units) y = cos(y). At first, in a 'scientific' calculator instrument or in such a calculator program within the computer, do opt for the radian units. As a starting value of y, choose the approximate value p/4 = 0.7854, knowing that cos(p/4) = 0.707 ~ p/4. Now take cosine of 0.7854, we get 0.7071. Again take its cosine: we get 0.7602. Continue taking cosine of the result similarly, and you would soon find the number converging to 0.7390851332. Just like the final set of orbitals mentioned above, this number also gives back itself (here, however, upon operation by the cosine function, not by the Hartree-Fock equation)!

2.3. The Orbital Energy Expression:

The energy e_i of the spin orbital u_i (this energy is also identical for the constituent spatial orbital, say the MO) is immediately obtainable from the above Hartree-Fock equation via left-multiplication of both sides by u_i* followed by integration over the 1st-electronic coordinates. Noting that u_i is normalised (so that <u_i(1)|u_i(1)> = 1), this would give:
   e_i =  ∫ u_i*(1)(1) u_i(1) dt₁      [i.e., <u_i(1)|(1)|u_i(1)>]

       =  ∫ u_i*(1)(1) u_i(1) dt₁ +  ∫ u_i*(1) [_j=1{j(1)j(1)}] u_i(1) dt₁ 

       =  <u_i(1)|(1)| u_i(1)>  + _j=1[ ∫ u_i*(1) j(1) u_i(1) dt₁ ∫ u_i*(1) j(1) u_i(1) dt₁]

Putting the above definitions for j (1) u_i (1) and j (1) u_i (1) in the above expression, we get:

     e_i =  < u_i(1)|(1)| u_i(1)> + 

   _j=1[ ∫ ∫ u_i*(1) u_j*(2) (1/ r₁₂) u_j(2) u_i(1) dt₁dt₂ ∫ ∫ u_i*(1) u_j*(2) (1/ r₁₂) u_i(2) u_j(1) dt₁dt₂ ]

    Or, in other words, e_i =  h_ii + _j=1(J_ij K_ij)

    where the one-electron integral h_ii and the two-electron integrals J_ij & K_ij are defined as: 

            The core integral h_ii = < u_i(1)|(1)| u_i(1)>   

           the Coulomb integral J_ij = ∫ ∫ u_i*(1) u_j*(2) (1/ r₁₂) u_j(2) u_i(1) dt₁dt₂  

           and the exchange integral K_ij = ∫ ∫ u_i*(1) u_j*(2) (1/ r₁₂) u_i(2) u_j(1) dt₁dt₂ 

Notes: (i) There is a special notation for various two-electron integrals arising in this theory. As per this
notation, the integral ∫ ∫ u_a*(1) u_b(1) (1/ r₁₂) u_c*(2) u_d(2) dt₁dt₂ is denoted in the abbreviated form 
(ab | cd). Accordingly, J_ij = (ii | jj) whereas K_ij = (ij | ji), which implies that  

e_i =  h_ii + _j=1[(ii | jj)(ij | ji)]. Please do verify these representations for J_ij & K_ij yourself.

(ii) Additionally, using the concept of electron indistinguishability (implying that the 1st and the 2nd electronic coordinates within their aforesaid expressions are mutually interchangeable), we also find that J_ij = J_ji and K_ij = K_ji (please verify this).

In other words, the two subscripts within a Coulomb or an exchange integral are freely interchangeable! 

 

2.4. Derivation Hints and the Molecular Electronic Energy Expression:

Assuming Y_el to be an Slater determinant D of the occupied SOs, we find that the variation theorem requires the variational integral I_v = <Y_el |Ĥ_e| Y_el> = <D|Ĥ_e|D> to be a minimum (here Ĥ_e is the molecular electronic Hamiltonian equalling T_e + V_ee + V_eN + V_NN). As D depends on the SO functions, I_v is clearly a functional of these SO functions. Now, I_v may be minimised under the constraint of keeping the set of SOs orthonormal by employing the Lagrange's undetermined multipliers method (see Sherrill 2000). This procedure results in the aforesaid Hartree-Fock equation (Section 2.2) defining the SOs as its solution, so that I_v becomes a minimum. It also gives that the minimised I_v, i.e., the molecular electronic energy U_HF (as per this HF theory) is (note that this energy includes the internuclear repulsion V_NN): 
       U_HF  =  _i=1h_ii + ½ _i=1j=1(J_ijK_ij) + V_NN

Note: (i) It is difficult to miss the obvious similarities between the original (i.e., source) electronic Hamiltonian Ĥ_e and the

resulting U_HF = <D|Ĥ_e|D> expression. Noting the expanded forms {_i=1(– ½) and _i=1aå (–A_a/r_ia) respectively} for T_e and V_eN components of Ĥ_e, it is immediately obvious that 
T_e + V_eN =  _i=1(i), comparable to the resulting _i=1h_ii component in U_HF. Similarly, we see that the 
½ _i=1j=1(J_ij K_ij) component in U_HF may also be written as 

½ _i=1j=1 (J_ijK_ij)  as for all i = j cases (J_ij K_ij) clearly vanishes (please verify this yourself). Now, 

½ _i=1j=1 (J_ijK_ij) obviously equals _i=1j>i(J_ijK_ij), which expression is comparable to the 

V_ee = _i=1j>i (1/r_ij) component in Ĥ_e, noting further that every (1/r_ij) term (within the Ĥ_e operator) gets reflected as a 
(J_ij K_ij) term within the energy U_HF. The V_NN term in Ĥ_e, on the other hand, reappears unchanged with the U_HF expression. These comparisons, however, should never be extended further to mean any sort of equality between the compared components (except for V_NN): need to remember that Ĥ_e is an operator operating on the wavefunction Y_el, whereas U_HF  is the corresponding energy. 

(ii) Does this mean that there exists, according to the Hartree-Fock theory, a Schrodinger equation 

Ĥ_eD = U_HFD (i.e., Ĥ_eY_el = U_HFY_el)? Yes, it does exist, and from this equation only we have got 

the aforesaid relation U_HF = <D|Ĥ_e|D> (= <Y_el|Ĥ_e|Y_el>).

(iii) As discussed in note (i) above, U_HF may also be expressed as:  

       U_HF  =  _i=1h_ii + _i=1j>i(J_ijK_ij) + V_NN

However, the earlier expression is preferred for U_HF (used in Sherrill 2000), probably because that could be more easily related to the sum (_i=1e_i) of the occupied-orbital energies (as is done just below).

 

Let us now sum the electronic orbital energies e_i for all the n occupied orbitals (i.e., SOs) using the e_i expression in Section 2.3. This gives:

    _i=1e_i  =  _i=1{h_ii + _j=1(J_ijK_ij)}

                   =  _i=1h_ii + _i=1j=1(J_ijK_ij)

Does this sum equals the purely electronic molecular energy E_HF = U_HFV_NN (i.e., the one excluding the internuclear repulsion)? Obviously no, for we see above (in this Section) that 
U_HFV_NN  =  _i=1h_ii + ½ _i=1j=1(J_ij K_ij) which is clearly different from the above expression for _i=1e_i .

Rather we find that U_HFV_NN  =  _i=1e_i ½ _i=1j=1(J_ij K_ij). This means that the purely electronic molecular energy (U_HFV_NN, say E_HF) does not equal the sum of energies of all the occupied SOs (or, in other words, of all electrons), but rather it is less than that sum by the actual inter-electronic repulsion energy present in U_HF! This is because, when we sum the energies of the electrons individually, the inter-electronic repulsion terms would ultimately get counted twice. [As an example, the 1st-2nd electron repulsion would get counted first as 1st-2nd repulsion for the 1st electron, and then as 2nd-1st one for the 2nd electron.] Thus the molecule as a whole actually has only half the inter-electronic repulsions than the ones present in the sum of all the electronic energies! 

 

2.5. Koopmaan's Theorem and its Proof using Hartree-Fock Theory:

Koopmaan's theorem is stated as: The energy required to remove an electron from an orbital (say, an MO) equals the negative of that orbital energy. Though this theorem may look obvious or trivial to the inexperienced reader, it is not so. [It is not a trivial one, because the molecular electronic energy is not exactly equal to the sum of the energies of the occupied orbitals, as we have just found.] 

 

Let us consider a neutral molecule with n electrons. The Hartree-Fock molecular electronic energy U_HF (including internuclear repulsion V_NN) is given by (we intentionally use here this particular U_HF expression found in the last note of above section):

       U_HF  =  _i=1h_ii + _i=1j>i(J_ijK_ij) + V_NN 

Let an electron be removed from the k-th SO converting the molecule to a molecular cation. If the MOs and hence the SOs are assumed to remain same within the cation formed, then the expression for the cation's electronic energy U_HF+  is:

       U_HF+  =  {_i=1h_ii h_kk} + {_i=1j>i(J_ijK_ij) _i=1 (J_kiK_ki)} + V_NN

This is because in the U_HF+ expression, the components involving the k-th SO are missing. Thus, in case of the one-electron integrals, the missing term is h_kk, whereas in case of the two-electron integrals, the missing terms are the J_ij and K_ij ones having one of the subscript as k (as the two subscripts are interchangeable -- as was found in Section 2.3 notes -- k may be chosen as solely the 1st subscript). These J_ij and K_ij terms are nothing but the interaction expressions for the k-th SO with all the non k-th SO (if one participant is missing in a graduation ceremony, try to figure out how many, and also which, inter-person handshakes will be thus missing from the ceremony)! 

 

Thus we find that the energy difference for the cation and the neutral molecule is: 
U_HF+  U_HF = { h_kk +  _i=1 (J_kiK_ki)}

However, as is obvious from their definitions, J_kk = K_kk (verify this yourself) 

i.e., (J_kkK_kk) = 0, we have

_i=1 (J_kiK_ki) = _i=1 (J_kiK_ki) + (J_kkK_kk) = _i=1(J_kiK_ki) 

Thus  U_HF+  U_HF = {h_kk +  _i=1(J_kiK_ki)}  = e_k   (from definition of e_i in Section 2.3)

This means that the energy required (i.e., U_HF+ – U_HF) to remove an electron from the k-th SO, converting the molecule to a monopositive molecular cation, equals the negative (i.e., –e_k) of the orbital energy (e_k) of that k-th SO. Thus the Koopmaan's theorem has been proved.

 

2.6. The Spatial-Orbital Formulation of the Hartree-Fock Theory:

For closed-subshell molecular electronic configurations {e.g., the ground configuration (s_g1s)² (s_u*1s)² (s_g2s)² of Li₂, or (s_g1s)² for H₂}, all the occupied spatial (molecular) orbitals are doubly occupied, so that the two SOs (i.e., fa & fb) arising from any one MO (say, f) remain either both occupied or both unoccupied (depending on whether the source MO is occupied). In such situations we can express the orbital energy and the molecular electronic energy etc. in terms of integrals involving only the MOs instead of the SOs, thus providing a spatial-orbital formulation of the whole Hartree-Fock theory devoid of spin expressions. In doing so, we may note the following changes in the definitions and the expressions:

 

(i) J_ij gets defined as the integral  ∫ ∫f_i*(1) f_j*(2) (1/ r₁₂) f_j(2) f_i(1) dv₁dv₂ involving the MOs f_i & f_j (instead of the SOs u_i & u_j) and the purely spatial volume elements dv₁ and dv₂. However, as the spin-integral factor of the earlier-definition J_ij is always unity (as we know that  ∫a*(1)a(1)dw₁ = ∫b*(1)b(1) dw₁ =  ∫a*(2)a(2)dw₂ = ∫b*(2)b(2)dw₂ = 1), the J_ij value remains unchanged in the new definition. For any h_ii integral also, the spin-integral factor similarly remains unity, so that the h_ii values remain unchanged.

(ii) K_ij gets defined as the integral  ∫ ∫f_i*(1) f_j*(2) (1/ r₁₂) f_i(2) f_j(1) dv₁dv₂. However, if the two spin-functions (i.e., a or b functions) associated with the SOs u_i & u_j are different, the earlier-definition K_ij for such SOs would vanish (as  ∫a*(1)b(1)dw₁ = ∫b*(1)a(1)dw₁ =  ∫a*(2)b(2)dw₂ = ∫b*(2)a(2)dw₂ = 0). So in the expressions for energies etc., such K_ij terms should be made to remain absent. [In the earlier spin-orbital based definition, such K_ij would themselves vanish, but that would not automatically happen if this new, spatial-orbital based, definition of K_ij is used -- so they must here be kept explicitly absent]. On the other hand, if the two spin-functions associated with the two SOs are the same (i.e., both a or both b), the spin-integral factor becomes unity (as happened above in case of J_ij), and those K_ij values remain unchanged even in this new definition. 

(iii) For a single sum involving J_ij over various j, the sum over the n occupied SOs may be replaced by a sum over the n/2 occupied MOs, but as every MO gives two SOs of same energy, the J_ij term for the sum must be replaced with 2J_ij. Thus 

_j=1J_ij is to be replaced with  _j=1(2J_ij). However for the single sum over K_ij over various j, in case of a closed-subshell configuration, half of the K_ij terms vanish, as every j-th MO contributes only one non-vanishing K_ij (for which the underlying j-th SO spin-function is same as that for i-th SO). The non-vanishing K_ij terms however remain unchanged in value. Thus the single-sum  _j=1K_ij is to be replaced with the sum  _j=1K_ij. Thus we find that _j=1(J_ijK_ij) expression for the sum over the occupied SOs is to be replaced with the sum _j=1(2J_ijK_ij) over the occupied MOs. Thus we get the i-th MO energy as e_i =  h_ii + _j=1(2J_ij K_ij). 

(iv) For the closed-subshell configuration, the Fock operator (1) [in (1)f_i(1) = e_if_i(1)] is similarly definable as: 

(1) = (1) + _j=1{2j (1)j (1)} where j & j means:

j(1) f_i(1) = { ∫f_j^*(2) (1/ r₁₂) f_j(2) dv₂}f_i(1) and j(1) f_i(1) = { ∫f_j^*(2)(1/ r₁₂) f_i (2)dv₂}f_j (1)

The aforesaid expression e_i =  h_ii + _j=1(2J_ij K_ij) for the MO energies immediately follows from this definition of . To derive that expression, left-multiply both sides of the Hartree-Fock equation (1)f_i(1) = e_if_i(1) with f_i^*(1), and integrate over the 1st-electronic spatial coordinates. Similar to Section 2.3, this obviously gives 
e_i =  ∫f_i*(1)(1) f_i(1) dv₁. Please verify by yourself that this finally leads to the aforesaid expression for e_i.

 

Let us now sum the above-expressed MO energies e_i for all the n electrons occupying all the n/2 MOs. Each of these MOs is doubly occupied, and so here we need to sum (2e_i) over the n/2 occupied MOs:

    _i=1(2 e_i)  =  2 _i=1 e_i

                   =  2 _i=1h_ii + 2 _i=1j=1(2J_ijK_ij)

The molecular electronic energy U_HF here is: 
       U_HF  =  2 _i=1h_ii + _i=1j=1(2J_ijK_ij) + V_NN

 

Obviously this means that

       U_HF  =  2 _i=1e_{i i=1j=1}(2J_ijK_ij) + V_NN

implying (as in Section 2.4) that when one sums the energies of the electrons individually, the inter-electronic repulsion terms get counted twice and so the total inter-electronic repulsion expression [the double sum _i=1j=1(2J_ijK_ij)] must be subtracted from the MO-energies sum (i.e., 2._i=1e_i) to arrive at the purely-electronic molecular energy E_HF (i.e., U_HFV_NN).

 

2.7. The Roothan Equations -- A Practical Route to HF Computations:

The 

 

 

Bibliography:

 

1. Sherrill C.D., An Introduction to Hartree-Fock Molecular Orbital Theory <http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html>, June 2000.

 

2. Levine I.N., Quantum Chemistry, Prentice Hall of India, New Delhi (India), 2003.