General classification of theoretical chemistry approaches:

1. Post Hartree-Fock Methods in Quantum ChemistrySourav PalNational Chemical LaboratoryPune- 411 008

2. Classical Mechanics (CM) Quantum Mechanics (QM) Molecular Mechanics (MM) Ab initio methods Semi-empirical methods Density functional theory (DFT) General classification of theoretical chemistry approaches:

4. THEORETICAL MODEL CHEMISTRY • Should Include Electron Correlation ( two-electron repulsion effects) in an efficient manner. • Applicability Should be General • Results Should Scale Correctly with Number of Electrons N (Size) e.g. energy proportional to N • Dissociate into Fragments Correctly • Accuracy; Computationally Tractable ; Ab initio description

5. Electronic Structure Models • Hartree-Fock Method ( one-particle approximation) sufficient in many cases; amenable to simple interpretation of molecular orbital theory. The MO theory overweights the ionic parts and thus restricted HF fails to describe dissociation closed shell molecules into open fragments. • In many cases high level of electron correlation arising from two-electron repulsion needed calling for post-HF rigorous developments • Configuration interaction, perturbation theory and coupled-cluster methods are the methods of choice • Among these, coupled-cluster has emerged as a compact method to include correlation and describe size-dependence, dissociation correctly

6. What is electron correlation and why do we need it? F0 is a single determinantal wavefunction. Slater Determinant Recall that the SCF procedure accounts for electron-electron repulsion by optimizing the one-electron MOs in the presence of an average field of the other electrons. The result is that electrons in the same spatial MO (anti-parallel spins) are too close together. Their motion is actually correlated. Correlation of anti-parallel spins missing in Hartree-Fock theory Eel.cor. = Eexact - EHF (B.O. approx; non-relativistic H)

7. Electron Correlation • Instantaneous repulsion between electrons, missing in mean field or Hartree Fock method • Correlation between electrons of opposite spins, making them avoid each other • Virtual orbitals in Hartree-Fock method used for expanding the many-electron wave function in terms of configurations (determinants)

8. One way to see why simple MO theory does not work is that at dissociation, more than one determinant is important • Any post HF method, based on simple RHF method, may not work, in general. • Post-HF expansion must work on multi-determinant in such cases to correct the problem, in general. • The state-of-the-art rigorous method is multi-reference coupled-cluster theory, applicable to high accuracy results for any state, away from equilibrium (including dissociation), excited states etc. • Rigorous method for molecular interaction, properties and reactivity

9. Size Consistent and Size Extensive Size-consistent method - the energy of two molecules (or fragments) computed at large separation (100 Å) is equal to the twice energy of the individual molecule (fragment). Only defined if the molecules are non-interacting. EAB ( R AB) E A + E B Size-extensive method - the energy scales linearly with the number of particles. Full CI is size consistent and extensive. All forms of truncated CI are not. (Some forms of CI, esp. MR-CI are approximately size consistent and size extensive with a large enough reference space.)

10. RHF dissociation problem Consider H2 in a minimal basis composed of one atomic 1s orbital on each atom. Two AOs (c) leads to two MOs (f)…

11. The ground state wavefunction is: Slater determinant with two electrons in the bonding MO Expand the Slater Determinant Factor the spatial and spin parts H does not depend on spin Four terms in the AO basis Ionic terms, two electrons in one Atomic Orbital Covalent terms, two electrons shared between two AOs

12. H2 Potential Energy Surface H. + H. E At the dissociation limit, H2 must separate into two neutral atoms. 0 H H Bond stretching H H At the RHF level, the wavefunction, F, is 50% ionic and 50% covalent at all bond lengths. H2 does not dissociate correctly at the RHF level!! Should be 100% covalent at large internuclear separations.

14. RHF dissociation problem has several consequences: • Energies for stretched bonds are too large. Affects transition state structures - Ea are overestimated. • Equilibrium bond lengths are too short at the RHF level. (Potential well is too steep.) HF method ‘overbinds’ the molecule. • . • The wavefunction contains too much ‘ionic’ character; causing dipole moments (and also atomic charges) at the RHF level to be too large. However, SCF procedures recover ~99% of the total electronic energy around equilibrium. But, even for small molecules such as H2, the remaining fraction of the energy - the correlation energy - is ~110 kJ/mol, on the order of a chemical bond.

15. To overcome the RHF dissociation problem,Use two-configurational trial function that is a combination of F0 and F1 First, write a new wavefunction using the anti-bonding MO. The form is similar to F0, but describes an excited state: MO basis AO basis Ionic terms Covalent terms

16. Configuration interaction • Linear expansion of wave function in terms of determinants, classified as different ranks of hole-particle excited determinants • Matrix eigen-value equation • Very accurate determination of a few lowest eigenvalues using iterative techniques • Problem of proper scaling with size

17. Trial function- Linear combination of F0 and F1; two electron configurations. Ionic terms Covalent terms Three points: As the bond is displaced from equilibrium, the coefficients (a0, a1) vary until at large separations, a1 = -a0: Ionic terms disappear and the molecule dissociates correctly into two neutral atoms. The above wave function is an example of configuration interaction. The inclusion of anti-bonding character in the wavefunction allows the electrons to be further apart on average. Electronic motion is correlated. The electronic energy will be lower (two variational parameters).

18. Configuration Interaction Since the HF method yields the best single determinant wavefunction and provides about 99% of the total electronic energy, it is commonly used as the reference on which subsequent improvements are based. As a starting point, consider as a trial function a linear combination of Slater determinants: Multi-determinant wavefunction a0 is usually close to 1 (~0.9). • M basis functions yield M molecular orbitals. • For N electrons, N/2 orbitals are occupied in the RHF wavefunction. • M-N/2 are unoccupied or virtual (anti-bonding) orbitals.

19. Generate excited Slater determinants by promoting up to N electrons from the N/2 occupied to M-N/2 virtuals: b b b 9 a,b,c… = virtual MOs a a a a 8 a,b 7 c,d c 6 k,l k 5 i i,j i i i 4 3 i,j,k… = occupied MOs j j j 2 1 … Ref. Single Double Triple Quadruple Excitation level

20. Represent the space containing all N-fold excitations byY(N). Then the COMPLETE CI wavefunction has the form Where Linear combination of Slater determinants with single excitations Doubly excitations Triples N-fold excitation The complete YCI expanded in an infinite basis yields the exact solution to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.), often used as benchmark.

21. The various coefficients, , may be obtained in a variety of ways. A straightforward method is to use the Variation Principle. Expectation value of He. Energy is minimized wrt coeff In a fashion analogous to the HF eqns, the CI Schrodinger equation can be formulated as a matrix eigenvalue problem. The elements of the vector, , are the coefficients, And the eigenvalue, EK, approximates the energy of the Kth state. E1 = ECI for the lowest state of a given symmetry and spin. E2 = 1st excited state of the same symmetry and spin, and so on.

22. Configuration State Functions Consider a single excitation from the RHF reference. Both FRHF and F(1) have Sz=0, but F(1) is not an eigenfunction of S2. FRHF F(1) Linear combination of singly excited determinants is an eigenfunction of S2. Configuration State Function, CSF (Spin Adapted Configuration, SAC) Only CSFs that have the same multiplicity as the HF reference contribute to the correlation energy. Singlet CSF

23. Multi-configuration Self-consistent Field (MCSCF) 9 Carry out Full CIandorbital optimization within a small active space. Six-electron in six-orbital MCSCF is shown. Written as [6,6]CASSCF. Complete Active Space Self-consistent Field (CASSCF) 8 7 6 H2O MOs Why? To have a better description of the ground or excited state. Some molecules are not well-described by a single Slater determinant, e.g. O3. To describe bond breaking/formation; Transition States. Open-shell system, especially low-spin. Low lying energy level(s); mixing with the ground state produces a better description of the electronic state. … 5 4 3 2 1

24. MCSCF Features: In general, the goal is to provide a better description of the main features of the electronic structure before attempting to recover most of the correlation energy. Some correlation energy (static correlation energy) is recovered. (So called dynamic correlation energy is obtained through CI and other methods through a large N-particle basis.) The choice of active space - occupied and virtual orbitals - is not always obvious. (Chemical intuition and experience help.) Convergence may be poor. CASSCF wavefunctions serve as excellent reference state(s) to recover a larger fraction of the dynamical correlation energy. A CISD calculation from a [n,m]-CASSCF reference is termed Multi-Reference CISD (MR-CISD). With a suitable active space, MRCISD approaches Full CI in accuracy for a given basis even though it is not size-extensive or consistent.

25. Examples of compounds that require MCSCF for a qualitatively correct description. zwitterionic Singlet state of twisted ethene, biradical. biradical Transition State

26. Mœller-Plesset Perturbation Theory In perturbation theory, the solution to one problem is expressed in terms of another one solved previously. The perturbation should be small in some sense relative to the known problem. Hamiltonian with pert., l Unperturbed Hamiltonian As the perturbation is turned on, W (the energy) and Y change. Use a Taylor series expansion in l.

27. Unperturbed H is the sum over Fock operators  Moller-Plesset (MP) pert th. Perturbation is a two-electron operator when H0 is the Fock operator. With the choice of H0, the first contribution to the correlation energy comes from double excitations. Explicit formula for 2nd order Moller-Plesset perturbation theory, MP2.

28. Advantages of MP’n’ Pert. Th. • MP2 computations on moderate sized systems (~150 basis functions) require the same effort as HF. Scales as M5, but in practice much less. • Size-extensive (but not variational). Size-extensivity is important; there is no error bound for energy differences. In other words, the error remains relatively constant for different systems. • Recovers ~80-90% of the correlation energy. • Can be extended to 4th order: MP4(SDQ) and MP4(SDTQ). MP4(SDTQ) recovers ~95-98% of the correlation energy, but scales as M7. • Because the computational effort is significanly less than CISD and the size-extensivity, MP2 is a good method for including electron correlation.

29. Coupled Cluster Theory Perturbation methods add all types of corrections, e.g., S,D,T,Q,..to a given order (2nd, 3rd, 4th,…). Coupled cluster (CC) methods include all corrections of a given type to infinite order. The CC wavefunction takes on a different form: Coupled Cluster Wavefunction F0 is the HF solution Exponential operator generates excited Slater determinants Cluster Operator N is the number of electrons

30. CC Theory cont. The T-operator acting on the HF reference generates all ith excited Slater Determinants, e.g. doubles Fijab. Expansion coefficients are called amplitudes; equivalent to the ai’s in the general multi-determinant wavefunction. HF ref. singles doubles triples Quadruple excitations The way that Slater determinants are generated is rather different…

31. CC Theory cont. HF reference Singly excited states Connected doubles Dis-connected doubles Connected triples, ‘true’ triples ‘Product’ Triples, disconnected triples True quadruples - four electrons interacting Product quadruples - two noninteracting pairs Product quadruples, and so on.

32. CC Theory cont. If all cluster operators up to TN are included, the method yields energies that are essentially equivalent to Full CI. In practice, only the singles and doubles excitation operators are used forming the Coupled Cluster Singles and Doubles model (CCSD). The result is that triple and quadruple excitations also enter into the energy expression (not shown) via products of single and double amplitudes. It has been shown that the connected triples term, T3, can be important. It can be included perturbatively at a modest cost to yield the CCSD(T) model. With the inclusion of connected triples, the CCSD(T) model yields energies close to the Full CI in the given basis, a very accurate wavefunction.

33. Comparison of Models Accuracy with a medium sized basis set (single determinant reference): HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T) In cases where there is (a) strong multi-reference character and (b) for excited states, MR-CI methods may be the best option.

34. Size-dependence • Dimer of non-interacting H2 molecules • D-CI wave function contains doubly excited determinants on each H2 molecule , but not the quadruply excited determinant • The product of monomer D CI, however, contains the quadruply excited • Φab x Φcd • However, exponential wave function of dimer even at doubles level contains quadruply excited determinant

35. MOLECULAR PROPERTIES • Defined as derivatives of molecular energy with respect to perturbation parameters. EXAMPLES:DIPOLE MOMENT, POLARIZIBILITY, NUCLEAR GRADIENTS AND HESSIANS ETC., • Finite-field (numerical) method. • Numerical evaluation of energy derivatives by computing energy at least two different fields. • Evaluate (no extra effort other than calculating energy) • Very inaccurate as it involves taking difference of two large. numbers.

36. ANALYTIC METHOD • Closed analytic form for energy derivatives. • Requires extra effort to solve for energy • Response equations. • easier evaluation of energy and property surfaces. • Properties for variational (stationary) theories • S.Pal, TCA 66, 151(1984); PRA 34, 2682(1986); PRA 33, 2240(1986); TCA 68, 379(1985); Vaval, Ghose and Pal, JCP 101, 4914 (1994); Vaval & Pal, PRA 54, 250(1996).

37. Coupled-Cluster Response • H()=H+  O • Equations for different order response: • E (1) = <  | exp(-T) {O +[H,T (1) ]} exp (T)|  > • 0 = < * | exp(-T) {O+[H,T (1) ]} exp (T)|  > • Linear Equation to be solved for T (1) amplitudes • Due to multi-commutator expansion, extensivity of properties retained

38. Properties with coupled cluster method • Non-variational theory • No Hellmann-Feynman theorem • Energy first derivative depends on first derivatives of cluster amplitudes which have to be explicitly computed. E(1) = YT T(1) + …. y T,A (Perturbation independent) AT(1) = B B Perturbation independent

39. Z-VECTOR METHOD FOR SRCC • Recasting energy derivative expression to eliminate perturbation dependent cluster derivative in favour of perturbation independent z-vector • E(1) = yTA-1B = ZTB ( ZTset of de-excitation amplitudes)+.. • Z-vector needs to be solved only once. ZTA = yT (Perturbation independent) • Dalgarno and Stewart, Handy and Schaefer • Z-Vector can be introduced by stationary approach using Lagrange multipliers • £ = E -  ZiT< i* | exp(-T) H exp (T)|  > • £ /  T = 0 provides Z amplitudes same as non-var

40. Case of near-degeneracy • At away from equilibrium, more than one configuration is important (non-dynamic correlation) • Perturbative or cluster expansion around any one determinant causes convergence problem • Coupled-cluster equations based on single reference suffer from convergence (intruder) • A correct way to solve the problem is to start from the a space of important determinants and the exponential wave-operator to generate dynamic correlation • Several different versions of multi-reference CC theory • Hilbert and Fock space, State-selective type

41. Multi-reference coupled-cluster • For near-degenerate systems, methods with starting space consisting of a linear combination of multi-determinants and an exponential operator acting on this space constitute a class of multi-reference coupled-cluster theories • Very powerful method with proper combination of dynamic and non-dynamic electron correlation • The nature of T operator can vary in this case, depending on the nature of the MRCC method. • Multiple roots obtained by diagonalisation of an effective Hamiltonian over the starting model space • Contribution to the Fock space MRCC, ideal for ionization/ excitation energies etc. • (Mukherjee D and Pal S, Adv Quant Chem 20, 291 (1989); Pal et al, J. Chem. Phys. 88, 4357 (1988); Vaval and Pal, J. Chem. Phys. 111, 4051 (1999)

42. MULTI-REFERENCE THEORIES : EFFECTIVE HAMILTONIAN APPROACH • Define quasi-degenerate model space p • P° =  l> <  ||(0)  =  Ci |  • Transform Hamiltonian by ‘’ to obtain an effective Hamiltonian such that it has same eigen values as the real Hamiltonian. P° Heff P° = P°H  P° (Heff)ij Cj = ECi • Obtain energies of all interacting states in model space by diagonalizing effective Hamiltonian over small dimensional model space p

43. MULTI-REFERENCE THEORIES : EFFECTIVE HAMILTONIAN APPROACH • Bloch equation H  =  Heff • Coupled cluster anastz for wave operator  = exp(T) • P[H  -  Heff ] P = 0 Q[H  -  Heff ] P = 0 HeffC = C E • Multiple states at a time at a particular geometry

44. Variants of Multi-reference CC • Effective Hamiltonian theory: Effective Hamiltonian over the model space of principal determinants constructed and energies obtained as eigen values of the effective Hamiltonian • Valence-universal or Fock space: Suitable for difference energies ( Mukherjee, Kutzelnigg, Lindgren, Kaldor and others) Common vacuum concept; Wave-operator consists of hole-particle creation, but also destruction of active holes and particles contained in the model space • State-universal or Hilbert space: Suitable for the potential energy surface. Each determinant acts as a vacuum (Jeziorski and Monkhorst; Jeziorski and Paldus ; Balkova and Bartlett)

45. Multi- reference coupled cluster thus is more general and powerful electronic structure theory • To make the theory applicable to energy derivatives like properties or gradients, Hessians etc., it is important to develop linear response to the MRCC theory Eqn for Cluster amplitude derivative and Heff • =  / { P[H  -  Heff ] P} = 0  = /{ Q[H  -  Heff ] P} = 0 Eqn. For energy derivative and model space coefficient derivative • Heff(1) C + HeffC(1) = C (1) E + C E(1) • S. Pal, Phys. Rev A 39, 39, (1989); S. Pal, Int. J. Quantum Chem, 41, 443 (1992)

46. Fock Space Multi-reference Coupled-Cluster Approach • ( Mukherjee and Pal, Adv. Quant. Chem. 20, 291 ,1989) • N-electron RHF chosen as a vacuum, with respect to which holes and particles are defined. • Subdivision of holes and particles into active and inactive space, depending on model space • General model space with m-particles and n-holes  (0) (m,n) =  iC i i (m,n)

47. Fock space MRCC • P(k,1)[H  -  Heff ] P(k,1) = 0 k = 0, m; 1=0, nQ(k,1)[H  -  Heff ] P(k,1) = 0 k = 0, m; 1=0, n Heff is the effective Hamiltonian defined over the model space determinants. The eigen values of it gives the exact energies of interest. • For low-lying excited states one hole one particle model space is suitable.

48. Fock Space MRCC • For the general one active hole and one active particle problem the model space is written as|µ(0)(k,1)> = iCi |i(k,1)> k=0,1; l=0,1The wave operator will be {exp ( S(k,l)) } • S (k,1) = Ť(0,0) + Ť(0,1) + Ť(1,0) + Ť(1,1) • For the (1,1) problem, the model space is an incomplete model space (IMS). Though generally for IMS, to have linked cluster theorem, intermediate normalization has to be abandoned, for (1,1) model space, the equations can be derived assuming intermediate normalization.P(1,1) P(1,1) P(1,1) + P(1,1) T1(1,1) T1(0,0) P(1,1) +……

49. In addition, for (1,1) model space T1(1,1) operator is in the wave operator, but it does not contribute to the energy and thus can be neglected in the energy derivative or linear response problemComputationally full singles and doubles approximation has been used. For excitation energies closed part of the connected (H exp(T(0,0)) is dropped, to facilitate direct evaluation.Finally to get the singlet and triplet excited states one diagonalizes the spin integrated effective Hamiltonian matrices HSEE and HTEE

50. Z- Vector method for MRCC theory • In a compact form the response equation may be written as, • A T (1) = B • A : Perturbation -independent matrix • B : Perturbation-dependent column vector Two options: Eliminate T(1)in Heff(1) • Eliminate perturbation-dependent T(1) in energy expression