This implies that if we can evaluate integrals for s-type functions, all other integrals can by generated by differentiation. These mathematical advantages of GTOs overcome their physically bad behaviour and make them the most common type of basis function used in computational methods.
As the number of basis functions increases, the accuracy of the MOs improves. In the limit of a complete basis set (CBS),i.e.Nbasis→ ∞, the MOs become exact within the HF approximation and this is called theHartree-Fock limit. However, the number of one- and two-electron integrals grows rapidly with the size, Nbasis, of the basis set. The number of one-electron integral grows as N2
basis and the two-electron integrals as
N4
basis, which limitNbasis in practical calculation. However, as the size of the basis set is increased, the Variational Principle ensures that the results become better and the quality of a basis set can be assessed by running a calculation with an increasingly larger basis set.
2.7
Exact Wave Function and Correlation Energy
The Hartree-Fock and basis set approximations allow us to obtain, without any difficulty, an approximate wave function. However, we are always interested in the exact solution. Using two arguments, it is possible to obtain a general form of the exact wave function.
2.7.1 Assumptions
Suppose we have a complete set of spin orbitals {χi(x)} and that the exact wave function Ψexact is unknown.
Any wave function can be written as a linear combination of all possible Slater determinants formed from a complete set of spin orbitals{χi}.
All possible determinants can be described by referring to the Hartree-Fock de- terminant.
The first argument comes from the assumptions and the second uses excited determi- nants which are described below.
2.7.2 Excited Determinants
The Hartree-Fock method produces a set {χi} of Nbasis spin orbitals of which the Hartree-Fock ground state wave function contains only the firstN spin orbitals
|Ψ0i=|χ1χ2· · ·χaχb· · ·χNi, (2.87) which has the lowest energy (Variational Principle). The spin orbitals{χ1,· · · , χN}are called the occupied orbitals and {χN+1,· · · , χNbasis} the unoccupied or virtual. Since
we have Nbasis spin orbitals, it is possible to construct
Ndet= Nbasis N = Nbasis! N!(Nbasis−N)! (2.88)
different singly excited determinants. For example, if we replace an occupied orbital χa by an unoccupied orbitalχr, we form a single excited determinant
|Ψrai=|χ1χ2· · ·χrχb· · ·χNi. (2.89) The physical interpretation of (2.89) corresponds to the excitation of an electron in the orbital χa to virtual orbital χr. A doubly excited determinant is one in which two electrons have been promoted fromχa and χb toχr andχs
|Ψrsabi=|χ1χ2· · ·χrχs· · ·χNi. (2.90)
2.7.3 Exact Wave Function and Full CI
Since all Ndet determinants can be represented by either the Hartree-Fock ground state or singly, doubly, triply, · · ·,N-tuply excited states, the exact wave function has the form |Ψexacti=c0|Ψ0i+ X ar cr a|Ψrai+ X a<b r<s crs ab|Ψrsabi+ X a<b<c r<s<t crst abc Ψrstabc +· · · (2.91)
when the set {χi} is completed, i.e. Nbasis tends to infinity. This representation is called configuration interaction (CI) [38] as it is formed by different “configuration” of spin orbitals. Since it uses all the configurations it is call full-CI. Truncated-CI is presented in Section 2.8.1.
2.7. EXACT WAVE FUNCTION AND CORRELATION ENERGY 43
The lowest Hamiltonian expectation value over this exact wave function
Eexact=hΨexact|H¯|Ψexacti (2.92) is the exact non-relativistic ground state energy of the system within the Born-Oppenheimer approximation. The difference between this exact energy and the Hartree-Fock energy is called thecorrelation energy [39]
Ecorr=Eexact−EHF, (2.93)
and is a consequence of electrons with opposite spins not being correlated within the Hartree-Fock approximation.
Unfortunately, this procedure requires the use of an infinite basis which cannot be implemented. Even if one uses a finite basis, the number of determinants,Ndet, grows factorially with the number of spin orbitals. It is possible to apply a full-CI only to the smallest systems.
2.7.4 Electron Correlation
The Hartree-Fock method is one of the simplest methods to solve the Schr¨odinger equation, where the real electron-electron interaction is replaced by an average inter- action. With a sufficiently large basis set, the Hartree-Fock wave function is able to recover ∼ 99% of the total energy. Unfortunately, the remaining 1% is often very important for describing chemical phenomena such as bond formation or dissociation. The concept of electron correlation was first proposed by Wigner [39] in his study of free electrons in a metal and it was defined later by L¨owdin [40] as the difference between the exact and Hartree-Fock energies (2.93). Physically, it corresponds to the motion of the electrons being correlated. On average they are further apart than described by the Hartree-Fock wave function. Correlation energy is among the most important and difficult problems in quantum chemistry, and has been recently qualified as “the many-body problem at the heart of chemistry” [41].
Unlike the exact energy, the Hartree-Fock energy depends on the basis set size and whether one uses an RHF or UHF method. We clarified this point by taking as a reference the UHF energy with a complete basis
Eexact
Usually we do not know the exact energy, but we can sometimes compute the exact energy for a given basis set B, which yields the definition of the basis set correlation energy
EcorrB =EexactB −EUHFB . (2.95) Therefore, the problem of correlation energy becomes a two-dimensional problem as shown by Figure 2.2: the larger the one-electron expansion (basis set size) and the many-electron expansion (number of determinants), the better the results.
Infinite basis set
Full CI Increasing electron correlation
Incr
easing size basis set
HF CISD CISDT CISDTQ … 3-21G 6-31G(d) 6-31G(d,p) 6-311G(d,p) 6-311+G(df,pd) 6-311++G(3df,3pd) Exact solution Incr easing accuracy Incr
easing computational cost
Figure 2.2: Pople diagram showing the two-dimensional convergence of the exact solu- tion (adapted from Figure 1.1 of [42]).
Often when a problem is difficult, it is easier to decompose it into smaller problems. The correlation energy has been decomposed into many different parts. We briefly review here three possible decompositions.
2.7. EXACT WAVE FUNCTION AND CORRELATION ENERGY 45
2.7.4.1 Dynamic and Static Correlation
Thedynamic correlation is connected with the correlated motion of electrons which is not correctly described by HF theory [43]. It is responsible for lowering the energy by keeping electrons further apart. The remaining correlation energy is called static correlation and arises from the multi-determinantal nature of the wave function. It becomes important for systems away from the equilibrium geometry or with nearly degenerate excited states [41].
Ecorr=Estat+Edyn. (2.96)
2.7.4.2 Electron Correlation Pairs
Electron correlation comes from all combinations of electrons but electron pairs contribute the most to it, and it is possible to obtain a total correlation energy by summing all electron pair correlation energies Ecorr,ij [44–46]
Ecorr =
X
i<j
Ecorr,ij. (2.97)
As a general rule, electron pairs occupying a single orbital contribute the most to the total energy. Electron pairs with parallel spin (even in the same shell) have lower correlation energy, since the exchange effects between electrons of parallel spin are accounted for in the HF method.
2.7.4.3 Radial and Angular Correlation
In the special case of atoms, we can define the radial correlation energy as the difference between an “exact” wave function, expanded only withs-type functions, and the HF energy [47, 48]. The residual correlation energy is assigned to originate from
angular correlation. Radial correlation accounts for the tendency of electrons to be at different distances from the nucleus, whereas the angular correlation accounts for the tendency of two electrons to be on the opposite side of the nucleus: