Electron Correltion Methods HF method: electron-electron interction is replced by n verge interction E HF c E 0 E HF E 0 exct ground stte energy E HF HF energy for given bsis set HF Ec 0 - represents mesure for the error introduced by the HF pproximtion Dynmicl correltion relted to the movements of the individul electrons - short rnge effect - due the overestimtion of short-rnge electron repulsions in Hrtree- Fock wvefunctions Non-dynmicl correltion - relted to the fct tht in certin circumstnces the single reference ground stte SD wve-function is indequte to describe given moleculr stte (i.e. in the cse of ner degenercy between different configurtions) - long rnge effect Frnk Jensen, Introduction to Computtionl Chemistry, John Wiley nd Sons, New York, 1999
Correltion Energy: Is it importnt? 100 90 80 70 60 50 40 30 20 10 0 Totl electronic energy Correltion energy N 2 molecule: CE ~ 0.5% of the EE ~ 50% of the binding energy!
Long-rnge correltion wrong dissocition behvior for RHF wvefunction! Potentil energy curves for H 2 molecule
multideterminntl wve-function HF Ψ 0Ψ iψi i usully 0 1
Short-rnge correltion Consider the Hmiltonin for He tom: Close to r 12 = 0, the term 1/r 12 becomes infinite; however, the energy is finite. there must be n dditionl singulrity in the Hmiltonin which cncels 1/r 12 term for r 12 0. The only cndidte for this cnceling term is the kinetic energy. The RHF wvefunction overestimtes the probbility of finding the two electrons close together (the electrons re voided to get too close to ech other becuse the electrosttic interction is treted in only n verge mnner), nd this in turn implies n overestimte of the electron repulsion energy. This is dynmicl effect (relted to the electron movements) chrcteristic for short rnge distnces nd the corresponding energy is clled correltion energy. For electrons with prllel spins, the exchnge interction (Fermi hole) contributes significntly to the reduction of the overestimtion of 1/r 12 ; the electrons re lredy kept prt by the Puli principle, nd the effects of electron correltion neglect re firly minor.
Excited Slter Determinnts (ESD) Suppose we hve N electrons nd K bsis functions used to expnd the MOs ESD RHF formlism will give N/2 occupied MOs nd K-N/2 virtul MOs obtined by replcing MOs which re occupied in the HF determinnt by unoccupied MOs - singly, doubly, triply, qudruply, etc. excited reltive to the HF determinnt Totl number of ESD depends on the size of the bsis set If ll the possible ESD (in given bsis set) re included then ll the electron correltion energy is recovered
Methods including electron correltion re two-dimensionl!! In mny cses the interest is only in clculting the correltion energy ssocited with the vlence electrons Frozen Core Approximtion (FCA) = limiting the number of ESD to only those which cn be generted by exciting the vlence electrons - it is not justified in terms of totl energy becuse the correltion of core electrons gives substntil contribution. However, it is essentilly constnt fctor which drops out when reltive energies re clculted Methods for tking the electron correltion into ccount: Configurtion Interction (CI) Mny Body Perturbtion Theory (MBPT) Coupled Cluster (CC) Moller-Plesset (MP) Theory
Configurtion Interction (CI) -bsed on the vritionl principle, the tril wve-function being expressed s liner combintion of Slter determinnts The expnsion coefficients re determined by imposing tht the energy should be minimum. The MOs used for building the excited determinnts re tken from HF clcultion nd held fixed CI 0 SCF S S S D D D T T T... In the lrge bsis set limit, ll electron correltion methods scle t lest s K 5 Exmple Molecule: H 2 O Bsis set: 6-31G(d) => 19BF => 38 spin MOs (10 occupied, 28 virtul) The totl number of excited determinnts will be C 38 398637556 Mny of them will hve different spin multiplicity nd cn therefore be left out in the clcultion. Generting only the singlet Configurtionl Stte Functions (CSF) we still obtin 1002001 determinnts!!! Full CI method is only fesible for quite smll systems!!! 10
Configurtion Stte Functions Consider single excittion from the RHF reference. Both F RHF nd F (1) hve S z =0, but F (1) is not n eigenfunction of S 2. F RHF F (1) Liner combintion of singly excited determinnts is n eigenfunction of S 2. Configurtion Stte Function, CSF (Spin Adpted Configurtion, SAC) Singlet CSF Only CSFs tht hve the sme multiplicity s the HF reference F1,2 1 (1) 2 (2) 1 (2) 2 (1)
Truncted CI methods CI 0SCF SS DD T T... s Truncting the expnsion given bove t level one => CIS - CI with only single excited determinnts CID - CI with only doubly excited determinnts CISD - CI with Singles nd Doubles (scles s K 6 ) CISDT - CI with Singles, Doubles nd Triples (scles s K 8 ) CISDTQ - CI with Singles, Doubles, Triples nd Qudruples (scles s K 10 ) - gives results close to the full CI - cn only be pplied to smll molecules nd smll bsis sets D T CISD - the only CI method which is generlly fesible for lrge vriety of systems - recovers 80-90% of the vilble correltion energy
Multi-Configurtion Self-Consistent Field Method (MCSCF) - is the CI method in which the MOs re lso vried, long with the coefficients of the CI expnsion MCSCF methods - re minly used for generting qulittively correct wve-function - recover the sttic prt of the correltion (the energy lowering is due to the greter flexibility in the wve-function) dynmic correltion the correltion of the electrons motions In MCSCF methods the necessry configurtions must be selected CASSCF (Complete Active Spce SCF) - the selection of the configurtions is done by prtitioning the MOs into ctive nd inctive spces ctive MOs - some of the highest occupied nd some of the lowest unoccupied MOs Within the ctive MOs full CI is performed A more complete nottion for this kind of methods is: [n,m]-casscf - n electrons re distributed in ll possible wys in m orbitls
9 8 7 6 5 4 3 2 1 Crry out Full CI nd orbitl optimiztion within smll ctive spce. Six-electron in six-orbitl MCSCF is shown (written s [6,6]- CASSCF) Complete Active Spce Self-consistent Field (CASSCF) Why? 1. To hve better description of the ground or excited stte. Some molecules re not well-described by single Slter determinnt, e.g. O 3. 2. To describe bond breking/formtion; Trnsition Sttes. 3. Open-shell system, especilly low-spin. 4. Low lying energy level(s); mixing with the ground stte produces better description of the electronic stte. HF H 2 O MOs
Alterntive to CASSCF Restricted Active Spce SCF (RASSCF) RASSCF the ctive MOs re further divided into three sections: RAS1, RAS2 nd RAS3 RAS1 spce MOs doubly occupied in the HF reference determinnt RAS2 spce both occupied nd virtul MOs in the HF reference determinnt RAS3 spce MOs empty in the HF reference determinnt Configurtions in RAS2 re generted by full CI Additionl configurtions re generted by llowing for exmple mximum of two electrons to be excited from RAS1 nd mximum of two electrons to be excited to RAS3 RASSCF combines full CI in smll number of MOs (RAS2) nd CISD in lrger MO spce (RAS1 nd RAS3)
MØller-Plesset Perturbtion Theory - perturbtionl method in which the unperturbed Hmiltonin is chosen s sum over Fock opertors N N N N H 0 F i hi ( Jij Kij) hi 2 i1 i1 j1 i1 V ee The sum of Fock opertors counts the verge electron-electron repulsion twice nd the perturbtion is chosen the difference: Vee 2 V ee where V ee represents the exct opertor for the electron-electron repulsion It cn be shown (Jensen, pg.127) tht the zero order wve-function is the HF determinnt while the zero order energy is just the sum of MO energies. Also, the first order energy is exctly the HF energy so tht in this pproch the correltion energy is recovered strting with the second order correction (MP2 method) In ddition, the first contribution to the correltion energy involves sum over doubly excited determinnts which cn be generted by promoting two electrons from occupied MOs i nd j to virtul MOs nd b. The explicit formul for the second order Moller-Plesset correction is: E( MP2) occ vir F F i j F F F F i j b i j MP2 method - scles s K 5 - ccounts for cc. 80-90% of the correltion energy - is firly inexpensive (from the computtionl resources perspective) for systems with resonble number of bsis functions (100-200) b i j b F b F 2
Coupled Cluster (CC) Methods The ide in CC methods is to include ll corrections of given type to infinite order. The wve-function is written s: cc e where: T 0 e T 1 T T with the cluster opertor given by: T T1 T2 T3... T N 2... k0 1 T k! Acting on the HF reference wve function, the T i opertor genertes ll i-th excited Slter determinnts: T T 1 2... 0 0 occ i occ i j vir vir b t i t b ij i b ij The exponentil opertor my be rewritten s: T 1 2 1 3 e 1 T1 T2 T1 T3 T1 T2 T1... 2 6 First term genertes the reference HF wve-function Second term genertes ll singly excited determinnts First prentheses genertes ll doubly excited sttes (true doubly excited sttes by T 2 or product of singly excited sttes by the product T 1 T 1 k
The second prentheses genertes ll triply excited sttes, true (T 3 ) or products triples (T 1 T 2, T 1 T 1 T 1 ) The energy is given by: occ vir b b b t F F F F F F F F ij ti t j ti t j i j b i j b E E cc 0 i j b So, the coupled cluster correltion energy is determined completely by the singles nd doubles mplitudes nd the two-electron MO integrls Truncted Coupled Cluster Methods If ll T N opertors re included in T the CC wve-function is equivlent to full CI wvefunction, but this is possible only for the smllest systems. Trunction of T Including only the T 1 opertor there will be no improvement over HF, the lowest level of pproximtion being T=T 2 ( CCD=Coupled Cluster Doubles) If T=T 1 +T 2 CCSD scles s K 6 the only generlly pplicble model If T=T 1 +T 2 +T 3 CCSDT scles s K 8