Gaussian type orbitals (GTO) basis functions assume the radial part e αr2. Sc. cartesian GTO functions take the form

Transkriptio

1 QTMN, STO and GTO basis sets For an accurate, but easy presentation of molecular orbitals a good basis set is needed. In general, a complete basis consists of an infinite numer of basis functions, M =. The solution in a complete basis set is called as Hartree Fock limit and the difference from that is called as the basis-set truncation error. In the basis set Slater type orbitals (STO) the radial part is e ζr, where ζ is orbital exponent, see sec The infinite set {e ζr } ζ is complete, if ζ R, but in practice, only a limited number of ζ l are chosen by fitting to STO. STO is not very popular, because evaluation of two-electron integrals in STO is laborious. Gaussian type orbitals (GTO) basis functions assume the radial part e αr2. Sc. cartesian GTO functions take the form g ijk (r) = N x i y j z k e αr2, (9.20) where r = r q r c = x ^i + y ^j + z ^k, i, j and k are nonnegative integers, r c is position of the "center", usually the nucleus, and r q is position of the electron q. Now, l = i + j + k, and therefore, l = 0, 1, 2,... are s, p, d,... type GTO functions. If x i y j z k are replaced by spherical harmonics Y l ml, we have sc. "spherical gaussians" basis set. The size of the Fock matrix to be diagonalized can be reduced by contraction of the basis {g i } i to a smaller "contracted GTO basis" {χ j } j by QTMN, χ j = Σ i d ji g i, (9.21) where the contracted function χ j is a sum of primitive GTOfunctions g i. The coefficients d ji are determined by fitting χ j to atomic orbitals. Molecular orbitals are then written in the form Fig ψ i = Σ j c ji χ j for the coefficients c ji to be searched.

2 The contraction schemes of GTO are many, e.g.: minimal basis set DZ, double zeta basis set TZ, triple zeta basis set SV, split valence basis set DZP, double zeta basis set plus polarization STO NG, e.g. STO 3G (4s)/[2s], (9s5p)/[3s2p] 3 21G, 6 31G*, 6 31G** An incomplete basis set implies errors or deficiencies in the solution. One of these is sc. "basis set superposition error", which can be corrected by sc. "counterpoise correction". QTMN, Table 9.3. HF SCF energies (in units Hartree = ev = aj Table 9.4. HF SCF bondlengths (in units Bohr = Å) Electron correlation HF theory includes the Coulombin repulsion between electrons in an average way, in form of Hartree potential, only. This means, that the HF theory does not include the manybody effects or correlations. This is the "definition of correlations" in the first-principles methods Configuration state functions (CSF) Assuming the number of basis functions is n, then we have 2n spin-orbitals, which can be occupied with N electrons in different ways. Let us denote the ground state Slater determinant as Φ o and once excited determinant as Φ a p and twice excited one as Φ ab pq, etc. Now, the configuration state function (CSF) is any of these determinats or a linear combination of those, which is an eigenfunction of the hamiltonian and all operators commuting with the hamiltonian, e.g. the operator S 2. QTMN, Fig. 9.5.

3 9.6. Configuration interaction (CI) The exact N electron many-body wavefunction can be written as Ψ = C 0 Φ 0 + Σ a,p C a p Φ a p + Σ a<b,p<q C ab pq Φ ab pq + Σ a<b<c,p<q<r C abc pqr Φ abc pqr +..., i.e., as a linear combination of the CSFs defined above, and assuming that the one-electron basis set is complete. This means, that the CSFs or N-electron determinants form a complete CSF-basis for N-electron wavefunctions. The excat many-body wavefunction Ψ does not represent the one-electron picture of Hartree Fock theory, i.e., occupation configuration of one-electron orbitals, but instead, a superposition of those. Therefore, this is called configuration mixing or configuration interaction (CI). The concepts "full CI" and "basis set correlation energy" are definend in Fig Correlation phenomena can also be called structural/ static or dynamic depending on the interpretation of the case. QTMN, (9.23) Fig CI calculations Kertoimet C tilan Ψ CSF-sarjakehitelmään (9.31) saadaan samoin kuin edellä kertoimet c spin-orbitaalin ψ i sarjakehitelmään (9.14) diagonalisoimalla hamiltonin matriisi. Matriisielementeistä useat häviävät ja tärkeimmät CSF-kantafunktiot ovat Φ 0 ja kahdesti viritetyt Φ ab pq. Brillouinin teoreeman mukaan esim. Φ 0 H Φ a p = 0. Sen mukaan kuinka kehitelmä (9.31) katkaistaan (engl. limited CI) voidaan nimetä mm. seuraavia CI-tasoja: DCI SDCI SDTQCI Katkaistulta CI-menetelmältä puuttuu ns. "size-consistency" MCSCF and MRCI QTMN, CI-menetelmässä kaikki CSF-funktiot on "rakennettu" samoilla Φ 0 -konfiguraatiolle HF-menetelmällä optimoiduilla spin-orbitaaleilla. Jos myös kertoimet c kehitelmässä (9.14) ψ = Σ j Mc ji θ j optimoidaan samalla kun kertoimet C kehitelmässä (9.31), on kyseessä "multiconfiguration SCF" (MCSCF) -menetelmä. MCSCF-aaltofunktiossa ei välttämättä ole yhtä ainoaa peruskonfiguraatiota Φ 0, jota "parannetaan" viritetyillä tiloilla. "Complete active-space SCF" (CASSCF) on MCSCF-menetelmä, jossa yksi-elektroniorbitaalit jaetaan kolmeen ryhmään (inactive, active, virtual) sen mukaan kuinka niiden virityksiä otetaan mukaan CSF-tiloissa. "Multireference CI" (MRCI) on CI- ja MCSCF-menetelmien välimuoto, joka kuvaa hyvin kor-

4 9.9. Møller Plesset many-body perturbation theory Monihiukkashäiriöteoria (engl. many-body perturbation theory, MBPT) on toinen vaihtoehto HF-teorian parantamiseksi systemaattisella tavalla. Häiriöteoria ei nojaudu variaatioteoreemaan, mutta sen etuna on "size consistency". Møller Plesset-häiriöteoriassa valitaan vertailutilan operaattoriksi yksi elektroni-fock-operaattoreiden summa (9.27) H (0) = Σ i f i. HF-aaltofunktio Φ 0 on myös tämän operaattorin ominaisfunktio, ks. kirjan esim Valitaan 1. kertaluvun häiriöksi operaattori H (1) = H H (0), joka "korjaa" vertailutilan H (0) energian Hartree Fock-energiaksi, kun H on molekyylin hamiltonin operaattori (9.1b). Hartree Fock-energia on siis E HF = E (0) + E (1), missä ja E (0) = Φ 0 H (0) Φ 0 E (1) = Φ 0 H (1) Φ 0. Toisen kertaluvun korjaus on QTMN, (9.28) (9.29) missä Φ J ovat "viritettyjä" CSF-funktioita. Osoittajassa olevista matriisielementeistä häviävät kaikki muut paitsi ne, joissa Φ J on kahdesti virittynyt CSF. Tätä toisen kertaluvun MP-teoriaa Coupled-Cluster method Eräs tapa kirjoittaa korreloitunut N-elektroniaaltofunktio on Ψ = e C Ψ 0, missä Ψ 0 on Hartree-Fock-aaltofunktio e C = 1 + C + C 2 /2! + C 3 /3! +..., C on "klusterioperaattori" C = C 1 + C 2 + C C N ja C k tekee systeemiin k elektronista viritystä. Esimerkiksi C 1 Ψ 0 = Σ a,p t a pφ a p ja C 2 Ψ 0 = Σ a,b,p,q t ab pqφ ab pq. (9.31a) (9.31b) QTMN, (9.30a) (9.30b) Voidaan osoittaa, että esim. 2-elektro-nivirityksistä C 1 C 1 Ψ 0 ja C 2 Ψ 0 tulee ottaa huomioon vain jälkimmäinen, ns. "kytketty" (coupled), ks. diagrammit kuvassa 9.8. Sama koskee myös kaikkia muitakin virityksiä, josta menetelmän nimi seuraa. Niinpä esim. "couple cluster doubles" (CCD) approksimaatiossa C = C 2 ja Ψ e C 2 Ψ 0 ja Schrödingerin yhtälö kirjoitetaan H e C Ψ 0 = E e C Ψ 0. (9.32) Tuosta tulee ortogonaalisuusehtojen vuoksi energian lausekkeeksi E = E HF + Ψ 0 HC 2 Ψ 0. (9.33) Fig. 9.8.

5 Density functional theory (DFT) Density functional theory (DFT) is an alternative approach to solve the many-electron system Schrödinger equation (9.1). DFT is a natural approach for extended systems (solids, sizeable clusters or molecules), whereas the Hartree Fock wavefunction theory and its dervatives of are that for smaller systems: atoms and smaller molecules Hohenberg Kohn existence theorem The starting point is the electron density. All properties of the ground state system of electrons in a given external potential (e.g., of nuclei) uniquely depend on the electron density ρ(r). This is the first Hohenberg Kohn theorem. QTMN, Let us prove, that the ground state electron density uniquely gives its external potential, i.e., its hamiltonian, which proves the theorem. Thus, the theorem can be proven by assuming two different hamiltonians, which lead to the same ground state density: Hohenberg Kohn variational theorem The first Hohenberg Kohn theorem implies, that with variation of the electron density the total energy can be minimized to that of the ground state, but not below. Thus, minimizing E[ρ] = T[ρ] + V ee [ρ] + ρ(r) v(r) dr = E HK [ρ] + ρ(r) v(r) dr with the condition δ { E[ρ] µ ρ(r) dr } = 0 we get where µ is the chemical potetntial Kohn Sham equations µ = v(r) + δe HK [ρ]/δρ(r), QTMN, (9.36) (9.37) Introducing one-electron orbitals of noninteracting electrons or sc. Kohn Sham orbitals ψ i the ground state total energy can be written as where ρ(r) = Σ i ψ i (r) 2. The first term is the kinetic energy, the second is the potential energy, the third is Hartree energy and the last one is sc. exchange and correlation energy. Thus, the energy is a functional of electron density, E[ρ]. The last term corrects the independent electrons energy to the interacting electrons energy.

6 QTMN, Application of variational principle to the total energy, silimarly to HF earlier, here leads to the Kohn Sham equations f ψ i = ε i ψ i, (9.39) where (9.42) and (9.41) If E xc [ρ] was known, the exchange and correlation potential V xc [ρ] could be found as its functional derivative. Thus, with DFT we can keep the one-electron picture, although we have all the correlations fully included. Therefore, interpretation of the Kohn Sham orbitals as quasi-electron states is different from the wavefunction theory. It can be shown, e.g., that the eigenenergy of the highest occupied Kohn Sham orbital is the first ionization energy, exactly! Historically, the DFT was preceded by the Thomas Fermi method, see sec. 7.18, where however, calculation of the kinetic energy without the one-electron picture is not simple. The sc. X α -method derived from the HF theory by Slater is also reminiscent of DFT or LDA, see the next sec. X α -method includes exchange energy as a functional of electron density. Also, hungarian Gáspár had presented similar suggestion before Slater Local-density approximation (LDA) For DFT calculations the exchange and correlation energy functional E xc [ρ] needs to be known for a given ρ(r). This functional is known very accurately for the homogeneous electron gas (HEG), which can be described with a single parameter ρ 0 or r s = (3 / 4πρ 0 ) 1/3. In fact, the energies per electron in HEG ε xc [ρ 0 ] = ε x [ρ 0 ] + ε c [ρ 0 ] are known and the functional is then E xc [ρ] = ρ(r) ε xc LDA (ρ(r)) dr, QTMN, (9.43a) where ε xc LDA (ρ(r)) = ε xc [ρ 0 ], when ρ 0 = ρ(r). Thus, at every position r the ε xc is approximated by that of the HEG, when ρ 0 = ρ(r). This is the LDA. The LDA can be expected to be viable for conduction electrons of metals, for example, but is has turned out to be very useful in many other cases as well and for molecules, in particular. In general, LDA can be expected to be viable, in cases where the exchange and correlation hole is localized around the electron. While HF approach is accurate for an one-electron system, e.g. hydrogen atom, the LDA is accurate for an infinite HEG. Between these extremes the structure dependent correlations need to be considered and HF is completed with CI, for example. So far, the best corrections to LDA are based on the "nonlocal" functionals of the form ε xc NL[ρ(r); ρ(r)]. Also, many kind of hybrids of HF and DFT have turned out to be useful. Jokingly, we can say that the hamiltonian of wavefunction theories is exact, but the resulting wavefunctions are not, whereas in case of DFT it is vice versa, the hamiltonian is an approximate (with the functional V xc [ρ]), but the resulting wavefunction is exact (for that hamiltonian, within numerical accuracy).

7 QTMN, QTMN,