II. Un panorama general
5. Los resultados: magnitud y formas de regularización y
5.2 Los programas de mejoramiento urbano a
The HF method typically accounts for ~99% of the total energy[47]. A significant limitation of the HF approach is the lack of electron correlation. Considering that chemistry generally happens with the valence electrons, errors in the final 1% of the energy could lead to significant changes in the nature of bonds molecules form.
The simplest and most straightforward approach accounting for electron correlation is to build the trial wave function from multiple Slater determinants of varying excitation. The Configuration Interaction (CI) method is oldest method to do so and is expressed as
ΨCI = a0ΦHF+ aSΦS + S
∑
aDΦD+ D∑
… = aiΦi i=0∑
(2.27)where the subscripts S, D, and i indicate Singly, Doubly, up to the ith excited Slater
determinant. Excited Slater determinants are built by moving electrons into unoccupied or virtual orbitals, where the amount of virtual orbitals available depends on the size of the basis set used. The advantage of the CI approach is that by including a large number of excited determinants, most of the correlation energy can be obtained. However, the number of possible excited Slater determinants increases factorially with the number of electrons and basis functions. Consider calculations for one H2O molecule with the 6-
31G(d) basis set up to the 10th exited Slater determinant. The 6-31G(d) basis includes 38
spin-orbitals, of which only 10 are occupied and 28 are unoccupied (i.e., 10 electrons in H2O, so 38 – 10 = 28 unoccupied or virtual orbitals). For this relatively small system the
number of excited Slater determinants is of the order of 30 million! For calculations of mercury molecules which can have several hundred electrons and basis functions, this approach is effectively ruled out.
A second class of methods accounting for electron correlation are based on the theoretical framework of Many Body Perturbation Theory (MBPT). The idea behind MBPT is that the true energy of the system differs from the approximate solution by small amount or a smalln. A common flavor of MBPT is Møller-Plesset perturbation theory (MPn) where the starting or zeroth order approximation is taken as the solution of the HF equations. The perturbed Hamiltonian equation can be written as:
H
! = H!0+
λ
V! (2.28)where H!0 is the HF Hamiltonian , λ is a dimensionless parameter, and V! is the
perturbed Hamiltonian. The total energy and exact wave function can be expressed as
E = E(0)
+λE(1)+λ2E(2)+λ3E(3)+… Ψ = Ψ +λ(1)Ψ(1)+λ2Ψ(2)+λ3Ψ(3)+…
(2.29)
where the superscripts (1), (2), (3), represent the 1st, 2nd, 3rd, …, and nth order
perturbations. Expanding the Schrödinger equation and collecting terms over λ leads to
λ0 :H!0Ψ0 = E(0)Ψ0 λ(1):H!0Ψ(1)+V!Ψ0 = E(0)Ψ(1)+ E(1)Ψ0 λ(2):H!0Ψ(1)+V!Ψ(1) = E(0)Ψ(2)+ E(1)Ψ(1)+ E(2)Ψ0 λ(n):H!0Ψ(n)+V!Ψ(n−1)= E(i)Ψ(n−i) i=0 n
∑
(2.29)Equation (2.29) shows how the 1st order perturbation energy depends on the 0th order
energy, the 2nd order depends on the 1st, and so on. Although MPn theory can be used to
systematically increase the accuracy of the calculated correlation energy, there is no guarantee that the calculations will converge to a finite value. For systems that are multi- reference, such as molecules with degenerate ground states, MPn calculations tend to
give erroneous results[47]. In general, the MPn methods are only used up to the 4th order
perturbation[47]. For our mercury calculations we do not explicitly use the MPn family of methods, however, some methods such as CCSD(T), include MPn calculations.
Another family of ab initio methods accounting for electron correlation are the Coupled Cluster (CC) methods. The concept of CC methods is to include all corrections of a given type (Singlet, Doublet, Triplet, Quadruples, etc.) to an infinite order to the reference HF wavefunction via an excitation operator[47]. Mathematically the CC operator is expressed as
T! = T!1+ T!2+ T!3+…+T!N (2.30)
where T! is the excitation operator, T! is the operator for the Singly excited states, 1 T!2 for
the Doubly excited states, etc. The T!1 and T!2 can act of the reference HF wave function
as follows: T!1Φ0 = tiaΦia a vir
∑
i occ∑
T!2Φ0= tijabΦijab a<b vir∑
i< j occ∑
(2.31)where
Φ
0 is the HF reference wave function and t are the amplitudes. The amplitudes i
T!2Φ0 thus represent all the possible Singly and Doubly excited states of the reference
wave, where the number of states is limited by the size of the basis set. The CC wave function is defined as
ΨCC = e T!
Φ0 (2.32)
which then transforms the Schrödinger equation to
H !eT!
Φ0= ECCe T!
Φ0 (2.33)
Again, it is impossible to include an infinitely large T!N operator, which in theory would
provide the exact correlation energy. The CC methods are therefore usually truncated to some excitation level (i.e., S, D, T, or Q). For our mercury calculations, we used a hybrid truncated CC method with single and doublet excitations (CCSD) and a non-iterative triplet contribution evaluated by 4th order MBPT added to the CCSD results
(CCSD(T))[47]. Dubbed the “gold standard” as one of main ab initio methods accounting for electron correlation, CCSD(T) provides good accuracy at a moderate computational cost. Table 2.1 shows a comparison of the computational cost as a function of basis set for the CI, MPn, and CC methods.
Table 2.1 Comparison of CPU Scaling as a Function of Basis Set
Scaling Method CI MP CC Nbasis5 CIS MP2 Nbasis6 CISD MP3 CCSD Nbasis7 MP4 CCSD(T) Nbasis8 CISDT MP5 Nbasis9 MP6 Nbasis10 CISDTQ MP7 CCSDTQ
Note: Table is adapted from Jensen Source: [47].
In terms of accuracy the discussed methods follow the following trend[47]: HF << MP2 < CISD < CCSD < MP4 < CCSD(T) < CCSDTQ. For our calculations of mercury molecules the CCSD(T) method is one of the main methods we use.