Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU

1. CFM centro de física de materiales Daniel Sánchez Portal Ricardo Díez Muiño Centro de Física de Materiales Centro Mixto CSIC-UPV/EHU

2. Electronicstructurecalculations: Methodology and applicationstonanostructures Electroncorrelationmethods in Quantum Chemistry

3. Electronicstructurecalculations: Methodology and applicationstonanostructures Lectureson Quantum Chemistry: TuesdayMarch 17th: 9.45 --> 12.30 Theoreticalbackground WednesdayMarch 18th : 9.45 --> 12.30 Practicalexercise

4. Electronicstructurecalculations: Methodology and applicationstonanostructures • Outline • Briefintroduction • Hartree- Fock • Basis sets • ConfigurationInteraction • Many-bodyperturbationtheory • Coupled-clustermethods

5. Electronicstructurecalculations: Methodology and applicationstonanostructures • Outline • Briefintroduction • Hartree- Fock • Basis sets • ConfigurationInteraction • Many-bodyperturbationtheory • Coupled-clustermethods Post Hartree Fock

6. Electronicstructurecalculations: Methodology and applicationstonanostructures Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry—an aberration which is happily almost impossible–it would occasion a rapid and widespread degeneration of that science. Auguste Comte, 1830.

7. Electronicstructurecalculations: Methodology and applicationstonanostructures In conclusion, I would like to emphasize my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory, for increasingly many facets of chemical information, is already at hand. There is only one obstacle, namely, that someone must pay for the computing time. Robert Mulliken. Nobel Prize address, 1966.

8. Electronicstructurecalculations: Methodology and applicationstonanostructures Quantum chemistry is quantum mechanics applied to the electrons in atoms and molecules. Used to determine: 1.- Structure of the molecule (bond lengths, angles) 2.- Electronic energy (bond energies, enthalpies of formation, etc) 3.- Spectra (electronic, vibrational, rotational, etc) 4.- Electrical properties (dipole moment, polarizability) 5.- Molecular orbitals and derived properties such as effective charges, bond orders. 6.- Barriers to reaction and other rate properties.

9. Electronicstructurecalculations: Methodology and applicationstonanostructures Goal of quantum chemistrymethods: Multi-electronatoms and molecules Wave function of manyelectrons in anexternalpotential (Borh-Oppenheimer) Finitesystem (no periodicboundaryconditions)

10. Electronicstructurecalculations: Methodology and applicationstonanostructures Key quantity: The wave function • Fundamental object in quantum mechanics: wave function • We want to find special wave functions such that • where • This is a fundamentally many-body equation! • A large variety of methods have been proposed and are being used to solve this problem.

11. Electronicstructurecalculations: Methodology and applicationstonanostructures Variationalprinciple • This will be one of our main tools today. • It states that the energy calculated from an approximation to the true • wavefunction will always be greater than the true energy: • Thus, the better the wavefunction, the lower the energy. • At a minimum, the first derivative of the energy will be zero.

12. Electronicstructurecalculations: Methodology and applicationstonanostructures Quantum Chemistry: Learning a new language CIST, MP2, CC, CSF, TZV, ...

13. Electronicstructurecalculations: Methodology and applicationstonanostructures First (simple) approach: Hartreeapproximation • As a first guess, one may try to write the many-electron wavefunction as • a product of one-electron spin-orbitalsji(ri,si): Animportantfeature of theHartreedescriptionisthattheprobability of findingoneelectron at a particular point in spaceisindependent of theprobability of findinganyotherelectron at thatpoint in space. Thus, duetotheindependentparticlemodel, themotion of theelectrons in theHartreeapproximationisuncorrelated.

14. Electronicstructurecalculations: Methodology and applicationstonanostructures Spin-orbitals • One-electron spin-orbitalsji(ri,si) are constructed as the product of a spatial orbital and a spin function. In general, they are molecular orbitals.

15. Electronicstructurecalculations: Methodology and applicationstonanostructures Hartreeenergy Applyingthevariationalprincipletotheabove wave function, one can findthe single-particleHartreeequations: self-consistentequations mean field

16. Electronicstructurecalculations: Methodology and applicationstonanostructures Antisymmetric wave function: Slaterdeterminant ThemainproblemwithHartree’s wave functionisthatitviolatesPauli´sprinciple. The wave function of fermionsmustbeantisymmetric and thereforetwofermionscannotbe in thesame quantum state. Slaterdeterminant

17. Electronicstructurecalculations: Methodology and applicationstonanostructures Antisymmetric wave function: Slaterdeterminant • Exchanging any two rows of a determinant (exchanging two electrons) • leads to a change in sign  antisymmetry. • Two electrons in the same quantum state  two identical rows  • the determinant is zero. Slaterdeterminant

18. Electronicstructurecalculations: Methodology and applicationstonanostructures Variationalprincipleon a Slaterdeterminant: Hartree-Fock Theminimization of theenergy <Y|H|Y> assumingthatthe wave functionYis a Slaterdeterminant leads totheHartree-Fockapproximation. Thecorresponding single-particleHartree-Fockequations are thefollowing:

19. Electronicstructurecalculations: Methodology and applicationstonanostructures Variationalprincipleon a Slaterdeterminant: Hartree-Fock Coulomb term J Exchange term K IdenticaltoHartree New exchangeterm Again, thisis a mean fieldself-consistentmodel.

20. Electronicstructurecalculations: Methodology and applicationstonanostructures Variationalprincipleon a Slaterdeterminant: Hartree-Fock Noticethataddingthetermi=j in thesumsmodifiesnothing: Itcancelsout.

21. Electronicstructurecalculations: Methodology and applicationstonanostructures Hartree-Fockequations: Fockoperator We can define thefollowingoperatorsbytheiractiononan orbital: and fromthemwe define theFockoperator: so thatthe HF equations can bewritten as:

22. Electronicstructurecalculations: Methodology and applicationstonanostructures RestrictedHartree-Fock (RHF) and UnrestrictedHartree-Fock (UHF) RHF: thespatialpart of theone-electron spin-orbitalsji(ri,si) isidenticalfor spin-up and spin-down (closed-shell) UHF: thespatialpart of theone-electron spin-orbitalsji(ri,si) dependonthe spin-orientation. Here, the wavefunction may be not a proper spin eigenfunction (spin contamination). Theenergy of a UHF wave functionisalwayslowerthan (orequalto) thecorresponding RHF wave function (thereis more flexibility in theformer).

23. Electronicstructurecalculations: Methodology and applicationstonanostructures Accuracy of HF: Spin issues Slater determinants are always eigenfunctions of Sz. However, they are not necessarily eigenfunctions of S2. For the general case there are always linear combinations of determinants that are eigenfunctions of Sz and S2 at the same time. Such spin-adapted linear combination of determinants (configurations) are needed to describe open-shell systems.

24. Electronicstructurecalculations: Methodology and applicationstonanostructures Accuracy of Hartree-Fock Hartree-Fockcalculationsoftenaccountfor ~99% of the total energy of thesystem. Theproblemisthattheremaining ~1% can determine thephysical and chemicalproperties of thesystem.

25. Electronicstructurecalculations: Methodology and applicationstonanostructures Accuracy of Hartree-Fock Hence, wehavetoimproveover HF: Howto do that?

26. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets Orbitals are usuallyexpanded in basis sets.

27. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets Slater-type orbitals (STOs) n,l,m (r,,) = Nn,l,m,Yl,m (,) rn-1 e-r are characterized by quantum numbers n, l, and m and exponents (which characterize the radial 'size' ). Slater-type orbitals are similar to Hydrogenicorbitals in the regions close to the nuclei. Specifically, they have a non-zero slope near the nucleus on which they are located d/dr(exp(-r))r=0 = - so they can have proper electron-nucleus cusps.

28. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets Cartesian Gaussian-type orbitals (GTOs) a,b,c (r,,) = N'a,b,c,xaybzc exp(-r2), are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents  which govern the radial 'size’. GTOs have zero slope near r=0 because d/dr(exp(-r2))r=0 = 0. The Coulomb cusp at the origin is not properly described. But, computationally, multi-center integrals are much more efficiently obtained.

29. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets To overcome the cusp weakness of GTO functions, it is common to combine two, three, or more GTOs, with combination coefficients that are fixed (and not treated as parameters), into new functions called contracted GTOs or CGTOs. However, it is not possible to correctly produce a cusp by combining any number of Gaussian functions because every Gaussian has a zero slope at r = 0 as shown here.

30. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets: whatto do withthesebuildingbricks Minimum basis set: the number of basis functions is equal to the number of core and valence electrons in the atom. Double zeta (DZ): there are twice as many basis functions as there are core and valence electrons. Triple zeta (TZ): there are three times as many basis functions as the number of core and valence electrons. Quadruple zeta (QZ), Pentuple Zeta (PZ or 5Z), etc. In any of them: split valence basis means that only the number of basis functions representing the valence electrons is increased. HCN molecule: DZ basisallowsfor differentbonding in differentdirections N C H

31. Electronicstructurecalculations: Methodology and applicationstonanostructures Basis sets: whatto do withthesebuildingbricks Polarization functions: a basis function with a higher component of angular momentum is added, p-functions to s-based orbitals, d-functions to p-based orbitals, etc. Double Polarization functions: basis functions with two higher components of angular momentum are added. For instance, double zeta with polarization (DZP), triple zeta plus double polarization (TZDP), etc. Polarization functions give angular flexibility in forming molecular orbitals between valence atomic orbitals. Polarization functions also allow for angular correlations in describing the correlated motions of electrons. N C H

32. Electronicstructurecalculations: Methodology and applicationstonanostructures Electroncorrelation Hartree-Fock is an approximation: It replaces the instantaneous electron-electron repulsion by an average repulsion term. Strictlyspeaking, electroncorrelationenergyisdefined as thedifferencebetweenthe HF energy and thelowestpossibleenergythatone can obtainwithin a givenbasis set. Physically, itcorrespondstothefactthat, onaverage, theelectrons are furtherapartthanthesituationdescribedbythe (R)HF wave function. A clearexample in RHF: electrons are paired in molecular orbitals and thespatialoverlapbetweentheorbitals of suchpair-electronsisexactlyone!

33. Electronicstructurecalculations: Methodology and applicationstonanostructures Electroncorrelationmethods: Post Hartree-Fock ToimproveoverHartree-Fock and includeelectron-correlation, theeasiestwayistostartfromtheHartree-Fockapproximation and ADD new things. Differentmethodologieswillbedefinedbythedifferentwaysto ‘add’ thingstoHartree-Fock. Typically, they fall into two classes: • Wavefunction expansion: The most common approaches are Configuration Interaction (CI) and Coupled-Cluster Methods (CC, CCSD). • Perturbation theory: The most common approach is Møller-Plesset (MP2 or MP4).

34. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants.

35. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants. wherewe are adding new Slaterdeterminantsthat are singly (Ys), doubly (Yd), triply (Yt), quadruply (Yq), etc. Excitedrelativetothe original HF determinant. Thesedeterminants are oftenreferredto as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.

36. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction Thesedeterminants are oftenreferredto as Singles (S), Doubles (D), Triples (T), Quadruples (Q), etc.

37. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction (CI) CI has many variants but is always based on the idea of expanding the wavefunction as a sum of Slater determinants. YCI=a0YHF+aSYS+aDYD+…=SaiYi Againwe use thevariationalprinciple and look fortheaicoefficientsthatmakeminimalthe wave functionenergy. Löwdin (1955): Complete CI gives exact wavefunction for the given atomic basis. For an infinite basis, it provides the exact solution. Orbitals are NOT reoptimized in CI!

38. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction (CI) YCI=a0YHF+aSYS+aDYD+…=SaiYi Structure of the CI matrix Brillouin’stheorem: Matrixelementsbetweenthe HF referencedeterminant and singlyexcitedstates are zero.

39. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction (CI) YCI=a0YHF+aSYS+aDYD+…=SaiYi In ordertodevelop a computationallytractablemodel, thenumber of exciteddeterminants in the CI expansionmustbereduced. Truncatingtheexpansion at one (Ys) doesnotimprovethe HF resultbecause of Brillouin’stheorem. Thelowest CI levelthatimprovesover HF is CI withDoubles (CID). Thenumber of singles ismuchlowerthanthenumber of Doubles. Therefore, including singles isnot a bigdeal: CI with Singles and Doubles (CISD). Alsowith Triples: CISDT. AlsowithQuadruples (CISDTQ).

40. Electronicstructurecalculations: Methodology and applicationstonanostructures ConfigurationInteraction (CI) YCI=a0YHF+aSYS+aDYD+…=SaiYi Thelowest CI levelthatimprovesover HF is CI withDoubles (CID). Weights of excitedconfiguration in the Ne atom. Doubleshavethehighestweight!

41. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. Let’s try toillustratehow CI accountsforelectroncorrelationtaking as anexamplethedissociation of thehydrogenmolecule H2 Taketwo 1s orbitals, one in each center of themolecule, cA and cB cA cB HF

42. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. Thebasisdeterminantsfor a full CI calculation are thefollowing: Double Single Single Single : triplet SZ=1 Single : triplet SZ=-1 F2+F3triplet SZ=0 F2-F3singlet

43. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. ThegroundstateF0 and thedoublyexcitedF1 can beexpanded in terms of theatomicorbitals: ionic covalent Now, ifweincreasethe bond lengthtowardsinfinity, the HF wave functionisstill a mixture of ionic and covalentcomponents and, in thedissociationlimitwillbe 50% H+H- and 50% H0H0. Thisistotallywrong!! Electroncorrelationismissing: electrons try toavoideachother!

44. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. We can solvethatbyusing full CI. The full CI matrix can beshowntobe: For1Sgsymmetryonlythesetermsmatter Thevariationalparametersallowustochoosethebestcombinationforeachbondingdistance. Forinstance, theioniccomponentdisappearsfor a1=-a0

45. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. Theproblem can alsobetreatedwith a UHF wave function. Althoughthe UHF wave functiondoesnotsolveeverything: spin contamination. We introduce a variationalparameterc in thedefinition of the molecular orbitals. Nowthey are differentfor spin-up and spin-down. ionic covalent butnowwehave anadditional tripletcomponent

46. Electronicstructurecalculations: Methodology and applicationstonanostructures Example: correlation in the H2dissociationproblem. Allthisisconspicuous in theenergydiagram:

47. Electronicstructurecalculations: Methodology and applicationstonanostructures Multi-Referencecalculations Foralmostdegeneratelevelsitis crucial tooptimizetheorbitals as well: Multi-ReferenceSelf-ConsistentField (MRSCF): a kind of CI in whichtheorbitals, as well as thecoefficients, are optimized. Configurations included in MCSCF are defined by the active space. Multi-ReferenceConfigurationInteraction (MRCI): A MRSCF functionischosen as reference. Singles, doubles, etc., are generatedout of allthedeterminantsthatenterthe MRSCF.

48. Electronicstructurecalculations: Methodology and applicationstonanostructures Active space in CI Reduced CI methods Idea: Not all determinants are equally important. Ansatz: Only allow excitations from a subset of orbitals into a subset of virtual orbitals (active space). Allow only a maximal number of excitations.

49. Electronicstructurecalculations: Methodology and applicationstonanostructures Many-bodyperturbationtheory In perturbationtheory, theHamiltoniansplitsinto: Perturbation, i.e., itseffectshouldbesmall! UnperturbedHamiltonian

50. Electronicstructurecalculations: Methodology and applicationstonanostructures Møller-PlessetPerturbationTheory First, letusremembertheHartree-Fockequations: These are theself-consistentequationsthatthe single-particle wave functionsshouldfulfilltoobtaintheminimumenergyfor a single-determinantmany-body wave function. WehavetheFockoperator, defined as: so thatthe HF equations can bewritten as: