For a general case:

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ΨSD =1N!

χ1 1( ) χ2 1( ) L χN 1( )χ1 2( ) χ2 2( ) L χN 2( )

M M O M

χ1 N( ) χ2 N( ) L χN N( )

N – the total number of electrons, χ - a spin-orbital, a product of a spatial orbital and anelectron spin eigenfunction. More compactly

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ΨSD = χ1χ2χ3LχN

If two spin-orbitals differ only by the spin eigenfunction (together they represent a doublyfilled orbital) this is typically represented by writing the spatial wave function with asuperscript 2 to indicate double occupation. Thus, if χ1 and χ2 represented α and β spins inspatial orbital ψ1, one writes

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ΨSD = ψ12χ3LχN

Interesting properties of Slater determinants: every electron appears in every spin orbitalof the expansion – a manifestation of the fact that quantum particles are indistinguishable. Asecond feature – so-called quantum mechanical exchange.

functions. Similar to Hartree product orbitals, the HF MOs can be individually determined aseigenfunctions of a set of one-electron operators, but the interaction of each electron with thestatic field of all other electrons includes exchange effects on the Coulomb repulsion. Later,Roothaan (1951) described matrix algebraic equations to carry out HF calculations using abasis set representation for the MOs. We represent MOs in their typical form for closed-shellsystems (all electrons spin-paired, two per occupied orbital) with wave unctions represented asa single Slater determinant.

The one-electron Fock operator is defined for each electron i:

€

fi =12∇i2 −

Zkrikk

nuclei

∑ +ViHF j{ }

The final term, the HF potential, is 2Ji – Ki, and the Ji and Ki operators are Coulomb andexchange operators defined so as to compute the Jij and Kij integrals. To determine the MOsusing the Roothaan approach, we follow the procedure analogous to that described for Hückeltheory. First, given a set of N basis functions, we solve the secular equations:

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F11 − ES11 F12 − ES12 L F1N − ES1NF21 − ES21 F22 − ES22 L F2N − ES2N

M M O M

FN1 − ESN1 FN2 − ESN 2 L FNN − ESNN

= 0

The Hartree-Fock Self-consistent Field MethodFock first proposed the extension of Hartree’s SCF procedure to Slater-determinant wave

and find its various roots Ej. The values for the matrix elements F and S are computed explicitly.S – overlap matrix elements:

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Sij = ϕiϕ jdr∫

For a general matrix element Fµν we compute

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Fµν = µ −12∇2 ν − Zk

k

nuclei

∑ µ1rikν + Pλσ µν |λσ( )− 12

µλ |νσ( )

λσ

∑

€

µ gν - one-electron integral, where g is some operator:

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µ gν = φµg∫ φνdr

For example, if g = 1/rik,

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µ gν = φµ1rik

∫ φνdr

(µν|λσ) – two-electron (four-index) integral:

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(µν |λσ ) = φµ 1( )∫∫ φν 1( ) 1r12φλ 2( )φσ 2( )dr 1( )dr 2( )

φµ and φν represent wave function of one electron and φλ and φσ the other. (µν|λσ) – Coulombintegral and (µλ|νσ) – exchange integral.The exchange integrals are preceded by a factor of 1/2 because they are limited to electrons ofthe same spin while Coulomb interactions are present for any combination of spins.

The final term in the equation for matrix element Fµν is weighted by elements of the‘density matrix’ P. This matrix describes the degree to which individual basis functionscontribute to the many-electron wave function – how energetically important the Coulomb andexchange integrals should be. If a basis function does not contribute in a significant way toany occupied MO, the integrals involving that basis function shouldnot be important:

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Pλσ = 2 aλiaσii

occupied

∑

aζi – the coefficients specifying the normalized contributions of basis function ζ to MO i. Thefactor of 2 – due to the fact that we are considering only singlet wave functions in which allorbitals are doubly occupied.

We need to know the orbital coefficients to form the density matrix that is used in theFock matrix elements, but the purpose of solving the secular equation is to determine thoseorbital coefficients. Just as in the Hartree method, the HF method follows a SCF procedure:first we guess the orbital coefficients and then we iterate to convergence. The full process isdescribed schematically by the flow chart (see the next page).

The chief chemical limitation of the Hartree-Fock theory – the one-electron nature of theFock operators – other than exchange, all electron correlation is ignored. Electron correlationmay be important for various molecular properties. From a practical standpoint, HF theory hadsome very challenging technical problems to early computational chemists. One problem –choice of a basis set. The LCAO approach using hydrogenic orbitals is attractive but this basisset requires numerical solution of the four-index integrals (in Fock matrix elements) – a verytedious process. The number of four-index integrals is huge: each index runs over the totalnumber of basis functions – N4 total integrals to be evaluated – this quartic scaling behaviorwith respect to basis-set size is the bottleneck in HF theory.

Two philosophies how to make further progress. The first one: the HF equations are verypowerful but still chemically flawed. Thus, other approximations may be introduced tosimplify their solution and possibly at the same time improve their accuracy (by someparameterization to reproduce key experimental quantities) – the motivation of so-called‘semiempirical’ MO theories. The second philosophy – HF theory is considered as a steppingstone on the way to exact solution of the Schrödinger equation, starting from it one can

develop more sophisticated methods (wave functions) to eventually approach the exactsolution – ‘ab initio’ MO theory.

Choose a basis set

Choose a molecular geometry q(0)

Compute and store all overlap,one-electron, and two-electronintegrals

Guess initial density matrix P(0)

Construct and solve Hartree-Fock secular equation

Construct density matrix fromoccupied MOs

Is new density matrxi P(n)

sufficiently similar to olddensity matrix P(n-1)?

Replace P(n-1) with P(n)

Optimize molecular geometry ?

Output data forunoptimized geometry

Does the current geometrysatisfy the optimizationcriteria?

Output data for optimizedgeometry

Choose new geometryaccording to optimizationalgorithm

yes

yes

yes

no

nono

Flaw Chart of the HF SCF procedure

Including Electron Correlation in Molecular Orbital Theory

Dynamical vs. Non-dynamical Electron CorrelationHow we might modify the HF wave function to obtain a lower electronic energy? By

the variational principle, such a construction would be a more accurate wave function. Wecannot do better than the HF wave function with a single determinant – one obvious choice isto construct a wave function as linear combination of multiple determinants:

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Ψ = c0ΨHF + c1Ψ1 + c2Ψ2 +K

where the coefficients c reflect the weight of each determinant in the expansion and alsoensure normalization.

For the majority of chemical species, the main error in the HF approximation derivesfrom ignoring the correlated motion of each electron with every other – ‘dynamicalcorrelation’ – refers to the dynamical character of the electron-electron interaction. For mostsystems the HF wave function dominates in the linear combination; even though thecorrelation energy may be large, it tends to be made up from a sum of individually smallcontributions from other determinants.

However, in some cases one of or more of the other determinants may have coefficientsof similar magnitude to that for the HF wave function.

In general, the number N of singlet CSFs that can be formed from the distribution of melectrons in n orbitals is

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N =n! n+1( )!

m2

! m2

+1

! n− m

2

! n− m

2+1

!

(4,4) – N = 20. If we include all possible arrangements of electrons in MCSCF, the method iscalled ‘complete active space’ SCF – CASSCF or CAS. N increases very quickly with theactive space size. CASSCF calculations of methanol (CH3OH) including all valenceelectrons: (14,12) – N = 169884 – very expensive task.Full Configuration Interaction

What if we carry out a CASSCF calculation for all electrons including all orbitals in thecomplete active space – ‘full configuration interaction’ or ‘full CI’. Within the choice ofbasis set, this is the best possible calculation that can be done, because it considers thecontribution of every possible CSF. Thus, a full CI with an infinite basis set is an ‘exact’solution of the Schrödinger equation (non-relativistic, Born-Oppenheimer, time-independent).

The number of CSFs in a full CI can be staggeringly large. The trouble is not thenumber of electrons, which is a constant, but the number of basis functions. Methanol withthe 6-31G(d) basis set – the total number of basis functions is 38 – (14,38) active space – wemust optimize 2.4×1013 expansion coefficients, and this is really a rather small basis set forchemical purposes.

Full CI calculations with large basis sets are usually carried out for only the smallest ofmolecules. In larger systems, the practical restriction to smaller basis sets makes full CIcalculations less chemically interesting, but, as an optimal limit, they permit evaluation of thequality of other methods for including electron correlation.

Configuration InteractionSingle-determinant reference

If we consider all possible excited configurations that can be generated from the HFdeterminant, we have a full CI, but such a calculation is too demanding to accomplish. Whatif we reduce the CI problem by allowing only a limited number of excitations? How manywe should include? Let’s rewrite our wave function:

€

Ψ = a0ΨHF + air

r

vir .

∑i

occ .

∑ Ψir + aij

rs

r<s

vir .

∑i< j

occ .

∑ Ψijrs +K

i and j are occupied MOs in the HF ‘reference’ wave function, r and s are virtual MOs in ΨHFand the additional CSFs are generated by exciting an electron from the occupied orbital(s)indicated by subscripts into to the virtual orbitals indicated by superscripts. Thus, the secondterm (after ΨHF) in the summation includes all possible single electronic excitations, and thethird includes all possible double excitations, etc.

If we do not have any problem with non-dynamical correlation, we do not need toreoptimize MOs in each determinant. Then, the problem is reduced to determining theexpansion coefficients for each excited CSF that is included.

The energies E of N different CI wave functions (corresponding to different variationallydetermined sets of coefficients) can be determined from the N roots of the CI secularequation:

€

H11 − E H12 L H1N

H21 H22 − E L H2N

M M O M

HN1 HN 2 L HNN − E

= 0

€

Hmn = Ψm H Ψn H – the Hamiltonian operatorA special case – the matrix element between the HF determinant and a singly excited CSF:

€

H1n = ΨHF H Ψir = φr F φi

F is the Fock operator and i and r are the occupied and virtual HF orbitals in the singleexcitation. Since these orbitals are eigenfunctions of the Fock operator, we have

€

φr F φi = εi φr φi = εiδirAll matrix elements between the HF determinants and singly excited determinants are zero –Brillouin theorem (1934).How we truncate the CI expansion?

1) CIS – only single excitations are included – no use for ground states but the simplestmethod (corresponds to HF) for excited states.

2) CID – only double excitations are included.3) CISD – single and double excitations are included – the most popular single-reference CI method.

Density Functional TheoryShouldn’t things be simpler than dealing with a wave function for a many-electron

system? Rather than working with a wave function, which has rather odd units of probabilitydensity to the one-half power, why can’t we work with some physical observable indetermining the energy (and possibly other properties) of a molecule? What observable weshould choose then?

The Hamiltonian depends only on the positions and atomic numbers of the nuclei andthe total number of electrons. The dependence on total number of electrons immediatelysuggests that a useful physical observable would be the electron density ρ, because,integrated over all space, it gives the total number of electrons:

€

N = ρ r( )∫ drBecause the nuclei are effectively point charges, their positions correspond to local maximain the electron density and these maxima are also cusps. For each nucleus A located at an

electron density maxima rA

€

∂ρ rA( )∂rA rA=0

= −2ZAρ rA( )

Z – the atomic number of A, rA – the radial distance from A

€

ρ – the spherically averaged electron densityTherefore, given a known density, one could form a Hamiltonian operator, solve the

Schrödinger equation, and determine the wave functions and energy eigenvalues:Density contains all the information we need!

Choose a basis set

Choose a molecular geometry q(0)

Compute and store all overlap,one-electron, and two-electronintegrals

Guess initial density matrix P(0)

Construct and solve Kohn-Sham secular equation

Construct density matrix fromoccupied KS MOs

Is new density matrxi P(n)

sufficiently similar to olddensity matrix P(n-1)?

Replace P(n-1) with P(n)

Optimize molecular geometry ?

Output data forunoptimized geometry

Does the current geometrysatisfy the optimizationcriteria?

Output data for optimizedgeometry

Choose new geometryaccording to optimizationalgorithm

yes

yes

yes

no

nono

Flow chart of the KS SCF procedure