DynamiccorrelationStaticcorrelationisimportantformoleculeswheretheground state is well described only with more than one(nearly-)degenerate determinant;- Hartree-Fock single Slater determinant;MCSCF (and CASSCF)treat the single determinant deficienciesof Hartree-Fock, but they don't do a very good job of handlingdynamic correlation that comes from instantaneous repulsionof electrons at all separations;- In the Hartree-Fock approximation, each electron sees theaverage density of all of the other electrons (mean-fieldapproximation lacks electron dynamics);
Dynamic correlation • Static correlation is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant; – Hartree-Fock single Slater determinant; • MCSCF (and CASSCF) treat the single determinant deficiencies of Hartree-Fock, but they don’t do a very good job of handling dynamic correlation that comes from instantaneous repulsion of electrons at all separations; – In the Hartree-Fock approximation, each electron sees the average density of all of the other electrons (mean-field approximation lacks electron dynamics); 7
GoalsforcorrelatedmethodsWell defined- Applicable to all molecules with no ad-hoc choices;- Can be used to construct model chemistries;Efficient-Notrestrictedtoverysmallsystems;Variational- Upper limittotheexactenergy;Size extensive- E(A+B) = E(A) + E(B)- Needed for properdescription of thermochemistry;.Hierarchy of costvs.accuracy; So that calculations can be systematically improved;8
Goals for correlated methods • Well defined – Applicable to all molecules with no ad-hoc choices; – Can be used to construct model chemistries; • Efficient – Not restricted to very small systems; • Variational – Upper limit to the exact energy; • Size extensive – E(A+B) = E(A) + E(B) – Needed for proper description of thermochemistry; • Hierarchy of cost vs. accuracy; – So that calculations can be systematically improved; 8
Configurationinteractionaeyalo..=o+Zt+Z+ZijkijabijkabciaY。 =|Φ ...Φ,/referencedeterminant(Hartree- Fock wavefunction)Ya -=l d .Φ--d.di+Φn Isingly excited determinant(excite occupied orbital Φ, to unoccupied orbital Φ.)Yab =l d..-d.i...j-pdj*.., I doubly excited determinant(d, →Φa, Φ, →)etc.Ifcarriedouttoall possibleexcitationstoall possibleorbitals,calledafullconfigurationiteration(fullci)model;Thiswavefunctionwouldbeexactwithinagivenbasis;9
Configuration interaction etc. ( , ) | | doubly excited determinant (excite occupied orbital to unoccupied orbital ) | | singly excited determinant (Hartree - Fock wavefunction) | | referencedeterminant 1 1 1 1 1 1 1 1 0 1 0 i a j b i a i j b j n ab i j i a i a i n a i n abc ijk ijkabc abc ijk ab i j ijab ab i j a i i a a i t t t → → = = = = + + + + − + − + − + 9 If carried out to all possible excitations to all possible orbitals, called a full configuration iteration (full CI) model; This wavefunction would be exact within a given basis;
ConfigurationinteractionDetermine Cl coefficients using the variational principle+-o+y++Z.iaijabijkabcminimize E = [y'HYdt / [?"Ydt with respect to tCiS-include all single excitations- Useful for excited states, but does not change orimprove the ground state;CisD-includeall singleanddoubleexcitations-Mostusefulforcorrelatingthegroundstate;-O2V2 determinants (O=number of occ.orb., V=number of unocc. orb.);CISDT-singles,doublesandtriples-Limitedto small molecules,caO3v3determinants;Full Cl-all possibleexcitations((O+V)!/o!v!)?determinants;Exactforagivenbasisset;limitedtoca.14electronsin14orbitals;10
• Determine CI coefficients using the variational principle • CIS – include all single excitations – Useful for excited states, but does not change or improve the ground state; • CISD – include all single and double excitations – Most useful for correlating the ground state; – O2V 2 determinants (O=number of occ. orb., V=number of unocc. orb.); • CISDT – singles, doubles and triples – Limited to small molecules, ca O3V 3 determinants; • Full CI – all possible excitations – ((O+V)!/O!V!)2 determinants; – Exact for a given basis set; – limited to ca. 14 electrons in 14 orbitals; Configuration interaction E d d t t t t abc ijk ijkabc abc ijk ab i j ijab ab i j a i i a a i / with respect to minimize ˆ * * 0 = = + + + + H 10
Configuration interactionHt=EtVery largeeigenvalue problem, can be solvediteratively; Only linear terms in the Cl coefficients;Upper bound to the exact energy (variational)Applicable to excited states;Gradients simpler than for non-variationalmethods;11
Configuration interaction • Very large eigenvalue problem, can be solved iteratively; • Only linear terms in the CI coefficients; • Upper bound to the exact energy (variational); • Applicable to excited states; • Gradients simpler than for non-variational methods; Ht = E t 11