Interacting Quantum Atoms (IQA) and the chemical bond

Transcript

1. Interacting Quantum Atoms & The Chemical Bond ´ Angel Mart´ ın Pend´ as angel@fluor.quimica.uniovi.es Departamento de Qu´ ımica F´ ısica y Anal´ ıtica, Universidad de Oviedo Spain c ´ Angel Mart´ ın Pend´ as, 2005 (1) Outline 1. Introduction. 2. (QTAM quick reference guide.) 3. Electron density & Atoms in Molecules. 4. QTAM as a theory of Atoms in Molecules. 5. Interacting Quantum Atoms (IQA) & Chemical Bonds. 6. Chemical Insights from IQA. 7. Examples. 8. Summary, Conclusions, Perspectives. c ´ Angel Mart´ ın Pend´ as, 2005 (2)
2. Introduction Introduction 1. Introduction. Quantum Chemistry is now near Dirac’s reductionist goal: • Theoretical (DFT) & Computational achievements allow us to reach chemical accuracy in everyday molecules. Problems, however, if chemical insight is regarded: 1. Wavefunction information. 2. Epistemology of emergent phenomena. Chemistry has a language defined before Quantum Mechanics. • Chemists envision entities in interaction • These entities live in 3D space, and are embodied with properties: bonds, transferability, characteristic energies and reactivities... We need interpretations!. How do we extract chemically meaningful information from Ψ? Ψ Chemistry c ´ Angel Mart´ ın Pend´ as, 2005 (3) Introduction Introduction Traditional Theories of the Chemical Bond: • Run inevitably parallel to computational schemes. (VB, MO). • Sooner or later they fall into a fragment picture • May stay at the qualitative level... • Or try a quantitative approach ⇒ Energy decomposition analyses. Within the MO paradigm, • Theory of orbital interactions (Fukui, Hoffman) ◦ Based on Perturbation Theory. ◦ Qualitative. ◦ Very influential. ◦ Coined chemical language. ◦ Runs into problems on going quantitative. [P, H0] = 0, Ψ0 not eigenfunctions of H0 c ´ Angel Mart´ ın Pend´ as, 2005 (4)
3. Introduction Introduction • Symmetry Adapted Perturbation Theory. ◦ Many flavors, many ways to handle the overcompleteness of Ψ0. ◦ Leads to terms coming from antisymmetry, the so–called exchange terms: Eint = E1 pol + E1 exch + E2 pol + E2 exch + ... E1 pol ⇒ classical electrostatic interaction energy of the interpenetrating isolated fragments. Generally < 0 Polarization terms are usually expanded at large interfragment distances in a multipolar expansion, only valid if the fragments are non–overlapping. Monopole–Monopole Monopole–Dipole Dipole–Dipole ... A R B R C non− overlapping overlapping R overlapping E1 exch contains both pure exchange and orthogonalization contributions. Usually > 0. Many methods share these ideas. Electrons invade other fragments. c ´ Angel Mart´ ın Pend´ as, 2005 (5) Introduction Introduction • Energy decompositions in Fock Space. Kitaura–Morokuma, RVS, ... ◦ Partition Fock space in the bases of the MO’s of the fragments. ◦ Several truncated diagonalizations provide different interaction contributions. F =                   FAA oo (esx) FAA ov (ind) FAB oo (ex ) FAB ov (ct) FAA vo (ind) FAA vv (esx) FAB vo (ct) FAB vv (ex ) FBA oo (ex ) FBA ov (ct) FBB oo (esx) FBB ov (ind) FBA vo (ct) FBA vv (ex ) FBB vo (ind) FBB vv (esx)                   ◦ Exchange is mixed-up with antisymmetry/orthogonality: Exchange–repulsion. ◦ Very popular years ago. ◦ The energetic contributions are chemically appealing, difficult to ascribe to a physical origin. ◦ Difficult in open–shell cases. AB oo mixing elstat+exchange. AA ov mixing Polarization of A. AB ov mixing Charge Transfer from A to B. c ´ Angel Mart´ ın Pend´ as, 2005 (6)
4. Introduction Introduction • Ziegler–Rauk like Energy Decompositions. ◦ Mixed scheme based on physical & chemical ideas. ◦ Molecule Formation Preparation + electrostatics + antisymmetrization + orbital interaction. ◦ Ψ0 A ⊕ Ψ0 B → Ψ0∗ A ⊕ Ψ0∗ B → AΨ0∗ A ⊕ Ψ0∗ B = Ψ0∗ AB → ΨAB ◦ Eint = Eprep + Eelstat + EP auli−rep + Eorb−int ◦ Antisymmetrization/Orthogonalization of the fragments ⇒ Kinetic energy. ◦ Particular orbital contributions may be isolated. ◦ May be applied to general open–shell cases. Similar situation in the VB paradigm. Criticism: • Generality, linkage to calculational procedure. • Dependence on the reference. • Use of fictitious intermediate states. • Exchange included in/mixed-up with orthogonality. c ´ Angel Mart´ ın Pend´ as, 2005 (7) Introduction Introduction Density and the chemical bond. As old as Quantum Chemistry. Properly put into context by Berlin (1950’s): Hellmann-Feynman theorem and charge redistributions. Binding & antibinding regions ..., build–up of density in the binding region. Difficult to generalize to polyatomics. It leads naturally to study Difference maps: ∆ρ = ρf − ρi Usually, ρi = A ρA (in vacuo) The reference problem is huge. c ´ Angel Mart´ ın Pend´ as, 2005 (8)
5. Introduction Introduction ∆ρ map for F2 . spherically averaged 2P F’s 2pz aligned F’s How to avoid the indefinitions? Look at derivatives. • Charge accummulation & charge depletion Topological Theories of the Chemical Bond. c ´ Angel Mart´ ın Pend´ as, 2005 (9) Introduction Introduction The topology induced by many scalar fields constructed from reduced density matrices carry chemical information 1. Take a scalar field f : D → R 2. Construct its gradient field: ∇f 3. Obtain its CPs, isolate local maxima (M) or minima (m). 4. Build their attraction or repulsion basins: D = M(m) DM(m) A number of them, according to the scalar studied: (ρ, ELF, ...) QTAM is based on the attraction basins of ρ. • Part of many standard chemistry curricula. Many advocates and detractors. • The topological method integrates: ◦ A theory without external references. ◦ An exhaustive partition of the 3D space. ◦ A method to obtain binary relations between chemical objects. c ´ Angel Mart´ ın Pend´ as, 2005 (10)
6. Introduction Introduction • The topological method achieves: ◦ A theory of bonding: The CP’s carry chemical information ◦ A theory of atoms in molecules, and groups of atoms or functional groups. ◦ An additive partition of observables into basin contributions: ˆ OΩ = Ω ˆ o(r)dr • Problems with a theory of binding. AIM: a quantitative theory of binding based on the density. • Living in the 3D space. • Free of references. • Providing an exact partition of the density. • Providing an exact partition of the energy. • Consistent with conventional wisdom. • Whose terms display a clear physical interpretation. c ´ Angel Mart´ ın Pend´ as, 2005 (11) Quick card QTAM 2. QTAM Quick Reference Guide General features of ρ. The stationary electron density is obtained from Ψ(x1, x2, . . . xN , R1, . . . RM ) as: ρ(r) = N s1 . . . dx2 . . . dxN . . . dR1 . . . dRM Ψ∗(x1 . . . xN , R1 . . . RM )Ψ(x1 . . . xN , R1 . . . RM ), x being the spatial and spinorial coordinates of electrons, and R the spatial coordinates of the nuclei. Under the BO approximateion, Ψ(x1, x2, . . . xN ; R), y ρ(r; R) = N s1 . . . dx2 . . . dxN Ψ∗(x1 . . . xN ; R)Ψ(x1 . . . xN ; R), ρ is observable. For instance, from elastic X–Ray scattering, I(k) ρ(r)e2πik·r dr 2 = |A(k)|2, where A is the Fourier transform of ρ. c ´ Angel Mart´ ın Pend´ as, 2005 (12)
7. Survey QTAM Analytical properties of ρ. Cusp theorem: ¯ ρ(r) = 1 4π ρ(r)senθdθdφ −2Zα = ∂ ln ¯ ρ(r) ∂|r − Rα| r=Rα . (1) Asymptotic behavior: l´ ım r→∞ ¯ ρ(r) r2 Ztotal−N+1 √ 2IP e−2r √ IP, Hoffmann-Ostenhof and Hoffmann-Ostenhof inequalities: − 1 2 ∇2ρ + (IP − Z r )ρ ≤ 0. Monotonicity and convexity are not assured. c ´ Angel Mart´ ın Pend´ as, 2005 (13) Survey QTAM Basic Morphology of ρ. HCN (HF/6-311G(p,d)) We rectify the ρ field. c ´ Angel Mart´ ın Pend´ as, 2005 (14)
8. Survey QTAM Topological analysis of ρ. In general, the number of attractors of the ρ field coincide in number and position with the nuclei. There will be, therefore, as many attraction basins as atoms in the system. LiF (HF/6-311G*) LiCl (HF/6-311G*) The basins of equal atoms under similar bonding regimes seem to be transferable. Critical points ∇ρ = 0 are important, and classified according to the eigenvalues of the Hessian of ρ: (3,-1) ≡ two negative, one positive curvatures. c ´ Angel Mart´ ın Pend´ as, 2005 (15) Survey QTAM Usual representations. (LiH HF/TZV) a) Projected relief diagrams. Nuclear plane Perpendicular plane c ´ Angel Mart´ ın Pend´ as, 2005 (16)
9. Survey QTAM b) Projected contour diagrams. Nuclear plane Perpendicular plane c ´ Angel Mart´ ın Pend´ as, 2005 (17) Survey QTAM c) Projected gradient field diagrams. Nuclear plane Perpendicular plane c ´ Angel Mart´ ın Pend´ as, 2005 (18)
10. Survey QTAM (d) Isosurfaces. ρ = 0.01 ρ = 0.02 ρ = 0.05 c ´ Angel Mart´ ın Pend´ as, 2005 (19) Survey QTAM Bond critical points. (3, −1) critical points are identified with chemical bonds. This is, in principle, an empirical assignment. However, they usually coincide with the objects defined in chemistry, even in rather strange situations. H2O c ´ Angel Mart´ ın Pend´ as, 2005 (20)
11. Survey QTAM B2H6. Ring points. B H H H x y z H B H H 2.511 bohr 3.345 bohr 96.5 121.8 o o 2.253 bohr B2H4 Plane B2H2 Plane c ´ Angel Mart´ ın Pend´ as, 2005 (21) Survey QTAM C8H8. Cage points. c ´ Angel Mart´ ın Pend´ as, 2005 (22)
12. Survey QTAM Properties of critical points • Many kinds of scalar and tensor properties calculated at the critical points are easy to correlate with chemically relevant concepts. ◦ ρb itself is a measure of bond strength for a given pair of bonded atoms. ◦ It turns to be small in ionic compounds and large in covalent. ◦ It correlates with bond order • The distances from nuclei to BCP’s are good indicators of atomic size. • The ratio of curvatures of the Hessian, ellipticity, has been used as a measure of something like π character of the bond. • etc... c ´ Angel Mart´ ın Pend´ as, 2005 (23) Survey QTAM General Features of ∇2ρ. If positive ⇒ Local Charge depletion. If negative ⇒ Local Charge concentration. Atomic densities are reasonably appproximated by a series of exponentially decreasing segments, so we should expect a negative and positive region in the laplacian for each of these segments. These are usually identified with the classical atomic shells. This is correct until Ca, which shows only 3 shells. Other functions (ELF,...) allow for a more perfect match. The (3,-3) points of ∇2ρ are associated with important charge concentrations. The most exterior ones are the valence charge concentrations (VCC). They may be associated with bonds (BVCC), or with lone pairs (LVCC). c ´ Angel Mart´ ın Pend´ as, 2005 (24)
13. Survey QTAM He (HF) Ar (HF) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 lnρ -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 lnρ -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ∇2 ρ r (u.a) K -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ∇2 ρ r (u.a) K L M c ´ Angel Mart´ ın Pend´ as, 2005 (25) Survey QTAM ∇2ρ suffers a restructuration upon molecule formation: Cl2 , NaCl HF//6-311G** Conc. K Depl K Conc. L Depl L Conc. M Depl M Conc. K Depl K Conc. L Depl L Conc. M Conc. K Depl K Conc. L c ´ Angel Mart´ ın Pend´ as, 2005 (26)
14. Survey QTAM ∇2ρ changes more noticeably if symmetry breaks: H2 O:HF//6-311G** • There are 4 tetrahedrally arranged VCC now. • Two BVCC’s and two lone pairs. • They correspond to the VSEPR model: The angle between the two BVCC’s is 103.8◦ and 138.6◦ for the lone pairs. c ´ Angel Mart´ ın Pend´ as, 2005 (27) Survey QTAM QTAM observables Since the partition of space is exhaustive, all observables may be obtained as a sum of basin contributions: ˆ O = Ω Ω ˆ o dr The existence of limiting surfaces introduces a number of surface integrals, with interesting meanings. Example: Nuclear attraction between electrons of basin A and the nucleus of basin B. V AB en = −ZB ΩA ρ(r) |r − RB | dr c ´ Angel Mart´ ın Pend´ as, 2005 (28)
15. Density AIM 3. Electron density & Atoms in Molecules. We want to partition the diagonal first order density: ρ(r) = ρA (r) + ρB (r) (1) Suppose we have such a method: • How? Using our preferred physical, chemical,... definition of atoms within molecules Let us turn to the energy. Using the NR Coulomb Hamiltonian: ˆ Hel = i ˆ hi + i>j 1 rij ⇒ E = Tr(ρ1 ˆ h) + 1 2 Tr(ρ2 r−1 12 ) + Vnn Problem: How do we partition ρ1 (r ; r), ρ2 (r1 , r2 ) consistently? Very old problem (i.e. Ruedenberg, 1960’s). We are after: • Using only the information in (1). • Obtaining physically sound components. c ´ Angel Mart´ ın Pend´ as, 2005 (29) Density AIM Li & Parr proposed a simple solution to this problem (mid 1980’s). Based on an “equal footing” treatment of electrons in the atoms within the molecules: • Electrons might not be able to distinguish if they are being considered part of an atom or of the molecule as a whole. We define the atomic weighting functions (AWF) wA : ρA (r) = wA (r)ρ(r); A wA (r) = 1 1. Partition of ρ1 (r ; r): • ρ1,A (r ; r) = wA (r)ρ1 (r ; r) −1 2 ∇2ρA (r ; r) ρA (r ; r) = −1 2 ∇2ρB (r ; r) ρB (r ; r) = −1 2 ∇2ρ(r ; r) ρ(r ; r) • In a DFT scheme, TA [ρA ] = wA T[ρ] c ´ Angel Mart´ ın Pend´ as, 2005 (30)
16. Density AIM 2. Partition of ρ2 (r1 , r2 ): • ρ2AB (r1 , r2 ) = wA (r1 )wB (r2 )ρ2 (r1 , r2 ) ρ2AA (r1 , r2 ) ρA (r1 )ρA (r2 ) = ρ2BB (r1 , r2 ) ρB (r1 )ρB (r2 ) = ρ2AB (r1 , r2 ) ρA (r1 )ρB (r2 ) = ρ2 (r1 , r2 ) ρ(r1 )ρ(r2 ) This leads to a partition of the energy based on a partition of the density . E = A TA + V AA ee + V AA en + A>B V AB en + V AB ne + V AB nn + V AB ee TA = dr wA (r)ˆ tρ1 (r ; r), V AB en = −ZB dr ρ1,A (r) |RB − r| V AA ee = dr1 dr2 ρ2AA (r1 , r2 ) r12 , V AB nn = ZA ZB RAB V AB ee = dr1 dr2 (ρ2AB (r1 , r2 ) + ρ2BA (r1 , r2 )) r12 c ´ Angel Mart´ ın Pend´ as, 2005 (31) Density AIM AWF’s include very general atomic partitions: • Interpenetrating: A B ρ ρ A B Localized. A B ρ ρ A B Delocalized. • Non-interpenetrating ⇒ 3D space partitioning. A B ρ ρ A B c ´ Angel Mart´ ın Pend´ as, 2005 (32)
17. Density AIM Many known interpenetrating partition formalisms may be recast into AWF form: • Becke wA (r) = PA(r) A PA(r) , PA (r) = B=A 1 2 [1 − h[h[...h k times (νAB )]]], h(x) = 3 2 x − 1 2 x3 νAB = µAB + aAB (1 − µ2 AB ), µAB = rA−rB RAB , aAB = − χ2−1 2χ , χ = RA RB . • Fern´ andez Rico et. al. (Maximally spherical densities) ρ(r) = i,j Pij φi(r) φj(r) = A i,j PA ij φi(r) φj(r) = A ρA (r) A PA ij = Pij, wA (r) = ρA(r) ρ(r) PA ij = Pij[mA (i) Θ(ζi − ζj) + mA (j) Θ(ζj − ζi)], mA (i) = 1(0) if φi is (is not) A-centered. ζi ≡ Orbital exponent of φi, Θ(x > 0) = 1, Θ(x < 0) = 0, Θ(x = 0) = 1 2 • Modified Hirshfeld ρ(r) = A a∈A B b∈B ρAB ab φA a (r) φB b (r) = A ρ0 A (r) + A=B ρ0 AB (r) ⇒ wA (r) = ρ0 A (r) A ρ0 A (r) . c ´ Angel Mart´ ın Pend´ as, 2005 (33) Density AIM 0.0 1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 w(r) w(r) in CO molecule (*) topological radii M. Hirshfeld Becke Fdez Rico Becke (*) c ´ Angel Mart´ ın Pend´ as, 2005 (34)
18. Density AIM 0.0 0.5 1.0 1.5 2.0 CO molecule Internuclear axis ρ tot , ρ C , ρ O M. Hirshfeld Becke Fdez Rico ρ tot Becke densities C=0.705 O=1.380 Topological radii Bragg−Slater radii C=0.651 O=0.465 c ´ Angel Mart´ ın Pend´ as, 2005 (35) Density AIM All known non-interpenetrating partitions may be recast into AWF form: • Quantum Theory of Atoms in Molecules (QTAM) (Bader) : wA (r) = 1(0) if r ∈ ΩA (r / ∈ ΩA ) . r ∈ ΩA iff r ∈ A ω − limit set of ∇ρ(r). • Voronoi deformation densities (VDD) : wA (r) = 1(0) if r ∈ ΩA (r / ∈ ΩA ) . r ∈ ΩA iff r ∈ A Voronoi Polyhedron. KCaF3 c ´ Angel Mart´ ın Pend´ as, 2005 (36)
19. Density AIM ... Or we may use variational criteria to determine wA . Which is the best atom within a molecule? Many minds, ... many answers. • One solution: Parr’s program: ◦ Minimal energetic cost (promotion). ◦ A0 + B0 CT −→ A + B Promotion −→ A∗ + B∗ Interaction −→ AB E0 A + E0 B ρ0 A , ρ0 B E1 − − − − − − − − − − − − − → NA N0 A (µA − µB )dN EA + EB ρA , ρB E2 − − − − − − − − − → Promotion E. E∗ A + E∗ B ρ∗ A , ρ∗ B − − − − − − − − → Interact. E. EAB ρ∗ A + ρ∗ B ◦ Minimize E1 + E2 : Since E0 A + E0 B is constant ⇒equivalent to minimizing E∗ A + E∗ B , the Self–Energy of the Atoms in the Molecule. ◦ Leads to an integral equation for wA . In a diatomic: ZB |r1 − RB |) − ZA |r1 − RA |) ρ(r1 ) + dr2 [4ωA (r2 ) − 2]ρ2 (r1 , r2 ) r12 = 0 c ´ Angel Mart´ ın Pend´ as, 2005 (37) Density AIM • It admits Analytical solutions in simple cases. ◦ i) No interelectronic repulsion. Eself is controlled by Vne : wA (r) =    1, ∀r such that ZA /r1A > ZB /r1B 0, ∀r such that ZA /r1A < ZB /r1B A B A B Homodiatomic ZA > ZB -In H+ 2 , the QTAM basins minimize Eself . -Leads to exhaustive, lo- calized atomic densities ◦ ii) No nuclear attraction. Eself is controlled by Vee : wA (r) = 1 2 . ρA = ρB = ρ/2. Delocalized solution. • General case: competence between localizing (Vne ) and delocalizing (Vee ) forces. Usually, the influence of the nuclei is dominating. c ´ Angel Mart´ ın Pend´ as, 2005 (38)
20. Density AIM • Example: N2 CAS[10,10]//TZV(d,f) Re calculation. (Minimization of Eself ≡ maximization of Eint ) Partition V AB ee 2V AB en No Ven 30.77 -151.57 No Vee 20.01 -43.88 Maximize nuclear attractions better than electron repulsions. • The expected general partition is not exhaustive but very localized. ρ A A B ρ B • This has been shown so in approximate solutions (H2 ) c ´ Angel Mart´ ın Pend´ as, 2005 (39) Density AIM • ... But the partition has problems in heterodiatomics. (particularly if large CT’s are expected) ◦ large CT’s (ionization energy≡ IP+EA) have a huge Eself penalty . ◦ A minimization of Eself would favor more or less neutral atomic partitions. ◦ A Least promoted atoms approach strays towards small CT’s. ◦ ... eppur si muove. A large scientific body supports high CT’s in very ionic compounds. ◦ Example: LiH, HF//6-311G(p,d) Re . Partition Eself QH QTAM -7.723 -0.891 Voronoi -7.807 -0.648 QTAM(HF) -7.838 -0.241 No Vee -7.798 0.533 No Ven -0.495 -1.000 Li H Li H LiH IAS HF IAS c ´ Angel Mart´ ın Pend´ as, 2005 (40)
21. QTAM AIM 4. Atoms in Molecules & QTAM. There are other “best variational atom” approaches. The AWF’s (ωA ’s) of QTAM minimize Group Energies . • In NR Quantum Mechanics, Ψ minimizes the Schr¨ odinger Lagrangian: I[Ψ] = d1, . . . , dN 2 2m i (∇i Ψ∗ · ∇i Ψ) + ˆ V + λ Ψ∗Ψ ◦ Its Euler–Lagrange equation is ˆ HΨ = EΨ • The restriction of the I functional to a 1–electron domain Ω is: I[Ω, Ψ] = Ω d1 d1 d2, . . . , dN 2 2m i (∇i Ψ∗ · ∇i Ψ) + ˆ V + λ Ψ∗Ψ • Minimizing I[Ω, Ψ] with respect to Ψ and Ω in a variational problem with: ◦ Free boundaries (Neumann) ◦ Moving boundaries. -Problem of Hermiticity of operators. -Theory of Open Quantum Systems c ´ Angel Mart´ ın Pend´ as, 2005 (41) QTAM AIM δI[Ω, Ψ] = − 2 4m ∂Ω dS1 · d1 {∇1 Ψ∗δΨ + Ψ∗δ∇1 Ψ} δj+c.c. + 2 4mN Ω d1∇2ρ For stationary systems: δI = 0 ⇐⇒    ˆ HΨ = EΨ. ∂Ω dS · ∇ρ = 0. -This admits Ω = R3. -Also QTAM basins! QTAM basins are solutions of the Schr¨ odinger variational problem. In R3, the total energy is minimized. What is minimized in general Ω’s ? I[A, Ψ] = TA + V AA ee + B V AB en + 1 2 B=A V AB ee = EA self + B=A V AB en + 1 2 V AB ee Eel = A I[A, Ψ]. I[A, Ψ] ≡ Electronic group energy . No problem with CT’s. Well founded theory of Interacting Quantum Atoms (IQA). • A particular case of Parr’s density consistent partitioning. c ´ Angel Mart´ ın Pend´ as, 2005 (42)
22. QTAM AIM All pairs of basins interact with each other. E = A EA self TA + V AA ee + V AA en + A>B EAB int V AB en + V AB ne + V AB nn + V AB ee Compatible with several theoretical schemes: • Parr’s, Relaxed pair–potential theory, ... The most interesting terms are V AB ee , coming from the partition of ρ2 . V AB ee = V AB,Coul ee + V AB,exch ee + V AB,corr ee • ρ2 = ρC 2 + ρX 2 + ρcorr 2 . (Coulomb+Exchange+Correlation). ◦ ρC 2 (r1 , r2 ) = ρ(r1 )ρ(r2 ) ◦ ρX 2 (r1 , r2 ) = −ρ1 (r1 ; r2 )ρ1 (r2 ; r1 ) (Fock–Dirac exchange). ◦ ρcorr 2 (r1 , r2 ) = Difference. ◦ ρcorr 2 + ρX 2 = ρxc 2 . Exchange–Correlation. • This definition of correlation is free of an external reference (` a la Hartree–Fock). • The exchange is pure . No mixture with orthogonality constraints. • Everything is contained in Ψ. No reference. c ´ Angel Mart´ ın Pend´ as, 2005 (43) QTAM AIM If binding with respect to a reference is needed, then Ebind = A EA self −EA,0 + A>B EAB int = A EA def + A>B EAB int This defines the Group Deformation Energies , EA def . All Classical interaction terms comprise the Classical Interaction Energy , V AB clas = V AB en + V BA en + V AB nn + V AB,Coul ee ⇒ EAB int = V AB clas + V AB xc Some properties: • EA def > 0 with respect to a grand–canonical reference with the same electron population. In homodiatomics, with respect to the isolated atom. • EAB clas > 0 for homodiatomics (theorem in electrostatics). • EAB clas has a completeley different meaning with interpenetrating densities. A B ρ ρ A B A B ρ ρ A B c ´ Angel Mart´ ın Pend´ as, 2005 (44)
23. QTAM AIM Computation of Vee interactions. How? to disentangle r1 and r2 in    r−1 12 (1) ρ2(r1, r2) (2) (1) We have developed an efficient technique to evaluate IAB IAB = ΩA dr1 ΩB dr2 r−1 12 f(r1) f(r2). There are two cases: • A = B: Laplace expansion. r−1 12 = ∞ l=0 4π 2l + 1 rl < rl+1 > +l m=−l Slm (ˆ r1)Slm (ˆ r2) • A = B: Bipolar expansion. r−1 12 = 4π ∞ l1m1 ∞ l2m2 (−1)l1 Sl1m1 (ˆ r1) Sl2m2 (ˆ r2) l1+l2 l3=|l1−l2| Vl1l2l3 (r1, r2, R) T l3 l1m1l2m2 ( ˆ R) A B R r r 2 1 r12 θ ϕ 1 1 ϕ θ 2 2 c ´ Angel Mart´ ın Pend´ as, 2005 (45) QTAM AIM • Use the QTAM AWF’s, ωA to transform basin integrations into R3 ones. • Define also RA lm (r) = Nl ˆ r Slm (ˆ r) ωAf(r)(r) dˆ r; Nl = 4π/(2l + 1). • Then, IAB = ∞ l1m1 ∞ l2m2 N−1 l1 N−1 l2 ∞ 0 RA l1m1 (r1 )r2 1 dr1 ∞ 0 RB l2m2 (r2 )r2 2 dr2 D(r1 , r2 ) • This is an O(N4) algorithm. The intrabasin integrations may be reduced to O(N3). • The RA lm grids are precomputed for all A, r, l, m, only for non–equivalent–by–symmetry basins. (2) ρ2 may be separated by a monadic diagonalization. • A general ρ2 may be written as: ρ2 (1 , 2 ; 1, 2) = n a,b,c,d λabcd ψ∗ a (1 )ψ∗ b (2 )ψc (1)ψd (2) c ´ Angel Mart´ ın Pend´ as, 2005 (46)
24. QTAM AIM • The symmetries of ρ2 , ρ2 (1 , 2 ; 1, 2) = ρ2 (2 , 1 ; 2, 1) = −ρ2 (1 , 2 ; 2, 1) = −ρ2 (2 , 1 ; 1, 2) = ρ∗ 2 (1, 2; 1 , 2 ). impose the constraints λabcd = λbadc = −λabdc = −λbacd = λ∗ cdab . • If the orbitals are real, ρ2 (1, 2) = n a≥b,c≥d abcd ψa (1)ψb (1)ψc (2)ψd (2). with symmetric in (a, b) and (c, d). • Now we may diagonalize in the (a, b) and (c, d) pairs: ρ2 (1, 2) = n a≥b dab fab (1)fab (2) ⇒ IAB • If we do not introduce the r12 factor in the ρ2 integrations we get the number of pairs of electrons lying in a basin, fAA, and the number of pairs of electrons shared betwen two basins, fAB. The latter is usually called the delocalization index. c ´ Angel Mart´ ın Pend´ as, 2005 (47) IQA AIM 5. Interacting Quantum Atoms (IQA) & Chemical Bonds. The Molecular Hydrogen ion (H+ 2 ). -0.4 -0.2 0.0 0.2 0.4 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 E/Eh RHH /a0 E V T Vnn Vne • Follows closely Slater’s. • EPR state at large R • Bonding density? • Repulsive wall from Vnn • No antisymmetry effects. • Vne ⇒ ρ behavior. • Forces at work: -Classical electrostatics. -Confinement. Within QM. • A chemical bond? A 1–electron bound state? c ´ Angel Mart´ ın Pend´ as, 2005 (48)
25. IQA AIM What does the IQA picture add to this? -0.4 -0.2 0.0 0.2 0.4 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 E/Eh R HH /a0 2TA 2Ven AA 2Edef A Eint Q10 A E • Physical insight. • A real space image of the effects of ρ redistributions. • Binding is a monocentric effect. • Edef < 0, Ebind at Re . Eint negligible, ex- cept at short distances, where it dominates the repulsive wall • Q10 good measure of ∆ρ. QA lm = ΩA dr ρrlSlm (θ, φ). • Entanglement mediated attraction. c ´ Angel Mart´ ın Pend´ as, 2005 (49) IQA AIM The H2 molecule I. The 1Σ+ g state FCI//TZV(p,d) calculations. True chemical bond. −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0 2 4 6 8 10 Ql A/e.a0 l , ∇2ρb /e.a0 −5 rHH /a0 Q1 A Q2 A ∇2ρb • The atomic dipole Q10 changes sign. • Its maximum almost coincides with ∇2ρ crossing the axis. • The atomic quadrupole Q20 < 0. • Many energetic contribu- tions follow the dipole. c ´ Angel Mart´ ın Pend´ as, 2005 (50)
26. IQA AIM −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 EA def EAB int fAB −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 Ebind (integr) Ebind (analyt) IQA analysis of Ebind • l´ ımR→∞ Eself = E(H) • Edef is very small (8 kcal/mol) at Re . • Edef increases steeply at short distances. • Binding is basically inte- raction. • Two deformed H atoms interact strongly. • Delocalization is large. c ´ Angel Mart´ ın Pend´ as, 2005 (51) IQA AIM −0.2 −0.1 0.0 0.1 0.2 0.3 0 2 4 6 8 10 E/Eh rHH /a0 ∆TA ∆Ven AA Q10 A −0.2 −0.1 0.0 0.1 0.2 0.3 0 2 4 6 8 10 E/Eh rHH /a0 ∆Vee AA EA def A B Self–Energy: A balance of large components. • Edef > 0: from a restricted variational principle. • TA follows Q10 , the atomic dipole. • Vne has an important con- tribution from the delocali- zed electrons from atom B • V AA ee = 0. A direct mea- sure of electron delocaliza- tion • Delocalization changes all energetic components. The total energetic change is very small. • The environment per- turbation ⇒ ∆Eeff ∆ ˆ V ρA,0 ⇒ ∆Eself 0. c ´ Angel Mart´ ın Pend´ as, 2005 (52)
27. IQA AIM −0.5 0.0 0.5 1.0 1.5 0 2 4 6 8 10 E/Eh rHH /a0 Vnn AB −Ven AB Vee AB −0.5 0.0 0.5 1.0 1.5 0 2 4 6 8 10 E/Eh rHH /a0 JAB Jlr AB −0.5 0.0 0.5 1.0 1.5 0 2 4 6 8 10 E/Eh rHH /a0 EAB int Interaction energy decomposition. • For R ≥ 4.5 All compo- nents basically converge to Coulombic 1/R behavior. • For R ≤ 4.5 the series diver- ges. Overlap (Delocalization) is felt. (Multipolar regime vs. short–range regime. • Vee < J. An important part of the interelectron repulsion has become monocentric . Not a big Eself penalty, a big decrease in Eint ⇒ signature of covalency. c ´ Angel Mart´ ın Pend´ as, 2005 (53) IQA AIM −0.4 −0.3 −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 Vcl AB Vxc AB VX AB VAB corr −0.4 −0.3 −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 Interaction energy decomposition II. • Vclas > 0. Small even at Re. At variance with other analysis. • Eint is the only sta- bilizing contribution. Interaction binded molecule • Eint = Vclas + Vxc. Vxc is the stabilization mecha- nism. Exchange–correlation binded molecule. • Vxc = VX + Vcorr: ◦ At R Re, V AB corr > 0, |VX | |Vcorr|. Exchange ≡ Resonance binding. ◦ At R Re, V AB corr < 0, and V AB corr → VX . Cannot se- parate exchange from corre- lation. Dissociation problem & Static correlation. c ´ Angel Mart´ ın Pend´ as, 2005 (54)
28. IQA AIM Some numbers at the exp. geometry: Prop. HF CAS[2,2] FCI QA 1 -0.1021 -0.1068 -0.1064 QQ 2 -0.3617 -0.3452 -0.3488 TA 0.5608 0.5805 0.5849 V AA en -1.2153 -1.2277 -1.2251 V AA ee 0.1979 0.1628 0.1532 JAA 0.3957 0.3994 0.3984 V AA xc -0.1979 -0.2366 -0.2452 V AA X -0.1979 -0.1988 -0.1967 V AA corr 0.0000 -0.0378 -0.0486 EA def 0.0432 0.0154 0.0128 V AB en -0.5974 -0.5975 -0.5975 V AB ee 0.2619 0.2993 0.2871 JAB 0.5236 0.5237 0.5236 V AB xc -0.2619 -0.2244 -0.2365 V AB X -0.2619 -0.2522 -0.2510 V AB corr 0.0000 0.0279 0.0145 EAB int -0.2193 -0.1820 -0.1942 • The sensitivity of magnitudes to calculation level is very dependent on the type of magnitude. ◦ Intraatomic: J (0.9 %, 0.2 %); Vne (1, 0.2);T (5, 1); Vxc (20, 4). Those related to xc mo- re sensitive, in general. However, VX (0.4, 1) relatively stable. ◦ Interatomic: Similar behavior. J stable, Vxc sensitive through correlation. • Correlation has different behavior for intra–, inte- ratomic magnitudes. Correlation stabilizes intra-, destabilizes inter-. • HF delocalizes too many electrons. • Eint (FCI) is about 120 kcal/mol. The resonance energy VX 160 kcal/mol. In agreement with che- mical wisdom Edef decreases on increasing elec- tron correlation. c ´ Angel Mart´ ın Pend´ as, 2005 (55) IQA AIM Comparison with other partitions. A B -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 Energy (hartree) R (H-H) (bohr) QTAM Eint =Vxc +Vclassic Ebind =Edef +Eint Eint Edef -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 Energy (hartree) R (H-H) (bohr) M. Hirshfeld Eint =Vxc +Vclassic Ebind =Edef +Eint Eint Edef -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 Energy (hartree) R (H-H) (bohr) Becke Eint =Vxc +Vclassic Ebind =Edef +Eint Eint Edef -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 8 Energy (hartree) R (H-H) (bohr) Fdez Rico Eint =Vxc +Vclassic Ebind =Edef +Eint Eint Edef c ´ Angel Mart´ ın Pend´ as, 2005 (56)
29. IQA AIM The H2 molecule II. The 3Σ+ u state FCI//TZV(p,d) calculations. −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 Ql A/e.a0 rHH /a0 S T fAB(S) fAB(T) −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 EA def (S) EA def (T) EAB int (S) EAB int (T) −0.2 −0.1 0.0 0.1 0 2 4 6 8 10 E/Eh rHH /a0 Ebind (S) Ebind (T) The triplet does not accummulate bonding charge. Delocalization is smaller. A much smaller, but stabilizing, Eint . Why? The triplet has a much larger deformation energy. Why? c ´ Angel Mart´ ın Pend´ as, 2005 (57) IQA AIM −0.2 −0.1 0.0 0.1 0.2 0 1 2 3 4 5 6 E/Eh rHH /a0 ∆TA(S) ∆TA(T) ∆Ven AA(S) ∆Ven AA(T) −0.2 −0.1 0.0 0.1 0.2 0 1 2 3 4 5 6 E/Eh rHH /a0 ∆Vee AA(S) ∆Vee AA(T) EA def (S) EA def (T) Self–Energy: A balance of large components. • Smaller Vee • Larger T up to Re . • Strange ∆Vne < 0 • Final origin: ρ2 (r1 , r1 ) = 0. • Pauli repulsion? c ´ Angel Mart´ ın Pend´ as, 2005 (58)
30. IQA AIM −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 E/Eh rHH /a0 Vcl AB(S) Vxc AB(S) Vee AB(S) −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 E/Eh rHH /a0 Vcl AB(T) Vxc AB(T) Vee AB(T) Interaction energy and ρ2 : • Smaller resonance due to charge separation. • Greater Vclas . • Changing Vee . Since ρ2 (r1 , r1 ) = 0, when |r1 − r2 | is large ρ2 must increase. This happens if R Re . • Clear in the intracular density. c ´ Angel Mart´ ın Pend´ as, 2005 (59) IQA AIM −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 Ql A/e.a0 l rLiH /a0 QLi Q1 H Q1 Li Charge Transfer. The LiH molecule. R = 10, 5.5, 3 bohr. Hook Ionization. Forward & backwards polarization. CAS[2,2]//6-311G* calculations. c ´ Angel Mart´ ın Pend´ as, 2005 (60)
31. IQA AIM −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 E/EH RLiH /a0 Edef Eint E Edef Li Edef H fLiH Binding energy Analysis: The reference. • Edef /neutral atoms 140 kcal/mol. • Edef /ions −12 kcal/mol. Negative due to incomplete ionization. • Everything has two regimes. Atomic, Ionic. • Edef due to ELi self . • ELi net −ELi+ just -5 kcal/mol. • fLiH describes beautifully the ionization. c ´ Angel Mart´ ın Pend´ as, 2005 (61) IQA AIM −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 ∆ E/EH ∆ N (H) ∆ N (Li) ∆EH ∆ELi Lim=0.192(0.196) Lim=−0.024(−0.028) µ = ∂E ∂N v Charge Transfer Analysis. DFT–like magnitudes? • Analogous behavior to that predicted by DFT. • Step–like between “integer” N energies. • Very Hard to deform closed shells. • Completely different µ’s for Li and H. • Access to hardness, Fukui functions? c ´ Angel Mart´ ın Pend´ as, 2005 (62)
32. IQA AIM −3.00 −2.00 −1.00 0.00 1.00 2.00 1 2 3 4 5 6 7 8 9 10 ∆ E/EH RLiH /a0 −NH ZLi /R Ven HLi −NLi ZH /R Ven LiH −3.00 −2.00 −1.00 0.00 1.00 2.00 1 2 3 4 5 6 7 8 9 10 ∆ E/EH RLiH /a0 Vee NH NLi / R J Eint −0.35 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 1 2 3 4 5 6 7 8 9 10 RLiH /a0 Eint Vclas Vcorr VX Vxc Interaction Energy. Classical Behavior All the interaction terms are very well represented by their classical expressions Vclas is 86 % Eint at Re! Electrostatically binded system. Vxc resonance. Two regimes with a plateau. Weakly correlated system. c ´ Angel Mart´ ın Pend´ as, 2005 (63) IQA AIM Comparison with other methods: LiH ionically dissociates to closed–shells: Easy to manipulate at HF//6-311G* R = 3.0 bohr. Kitaura–Morokuma Prop. Value Eelstat -0.3182 Eexch−rep +0.0694 EP ol -0.0066 ECT -0.0502 ECT (BSSE) -0.0344 Eint -0.2828 Eint(BSSE) -0.2624 -Nominal charge Eelstat . -CT decreases ionicity. -Eexch−rep small! SAPT (BSSE corrected) Prop. Value E10 elst -0.2830 E10 exch 0.0632 E20 ind -0.1367 E20 ex−in 0.0787 Ecorr -0.0016 Eint -0.2776 -Nominal charge Eelstat . -Non negligible Eind . -Rather good Eint! IQA Prop. Value Eint -0.2886 Exc -0.0368 Eclas -0.2517 Eclas (Q = 1) -0.2760 Edef 0.0066 Ebind -0.2820 Ebind (BSSE) -0.2615 -Perturbed charged Eelstat . -Eclas (Q = 1) within 5 kcal/mol with SAPT. -Vxc = VX < 0! -Edef very small. |Eclas − Ebind | < 20kcal/mol. c ´ Angel Mart´ ın Pend´ as, 2005 (64)
33. IQA AIM Polar bonds: the 1A1 H2 O molecule. CAS[6,5]//6-311G** calculation. • 3 Quantum Atoms, 2 bond paths; 2 bonded, 1 non–bonded interactions. • A summary of the topological properties: Prop. Value ρb 0.3896 ∇2ρb -2.7361 G 0.0897 Q(H) 0.5598 • Large density at BCP. • Large negative Laplacian. • Considerable CT. • Shared interaction. • Two non–bonded VCC’s c ´ Angel Mart´ ın Pend´ as, 2005 (65) IQA AIM What do we expect on physical grounds? O–H interactions with    large Exchange/Resonance contributions. large Electrostatic terms. H–H interaction with important classical repulsion. Property O H ∆TA 0.6349 -0.1060 ∆V AA en -6.5823 0.1187 ∆V AA ee 6.1149 0.0431 EA def 0.1675 0.1956 OH HH EAB int -0.4897 0.1239 V AB xc -0.1976 -0.0055 V AB clas -0.2921 0.1295 JAB lr diverges 0.1376 fAB 0.6148 0.0404 • A reference is difficult to find. • The H atom looses charge ⇒ ∆T < 0, ∆Ven > 0. • OH delocalization very large ⇒ ∆Vee > 0. • Eint for OH very large. 60/40 partition into classi- cal/QM • Vxc for OH similar to the single bond in H2 . • HH interaction dominated by classical, multipolar repulsion. • Negligible HH delocalization. • Clear partition into bonded & non–bonded interactions. c ´ Angel Mart´ ın Pend´ as, 2005 (66)
34. IQA Chemical insights 6. Chemical insights from IQA The Picture is very stable. N2 1Σ+ g CAS[10,10]//TZV(2d,f) -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 1.5 2.0 2.5 3.0 3.5 E/Eh RNN /a0 Fdez. Rico Edef Eint Ebind Exc Eclas -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 1.5 2.0 2.5 3.0 3.5 E/Eh RNN /a0 QTAM Edef Eint Ebind Exc Eclas Vclas (Re) = −250 kcal/mol. Unreasonable Edef Vclas (Re) = 130 kcal/mol. Edef (Re) = 38 kcal/mol. c ´ Angel Mart´ ın Pend´ as, 2005 (67) IQA Chemical insights 1st, 2nd period homodiatomics: CAS[Valence,10]//TZV(2d,f) except He2 ,FCI//cc-PVTZ; Ne2 , CAS[14,12]//TZV(2df); Re Dc e De e EA def Eint Vclas fAB ∆TA ∆V AA en ∆V AA ee H2 0.741 105.0 104.0 8.3 -122.4 26.2 0.845 54.8 -143.1 96.6 He2 2.875 0.0 0.0 0.3 -0.6 0.0 0.005 0.1 0.1 0.3 Li2 2.674 28.0 26.0 16.8 -61.2 0.8 0.835 9.8 -53.7 65.2 Be2 2.523 1.2 2.4 29.2 -57.2 2.2 0.589 0.8 -30.9 59.4 B2 1.601 73.0 71.0 49.3 -170.2 27.7 1.368 36.8 -245.7 258.2 C2 1.254 147.0 145.0 50.7 -247.9 83.8 1.803 69.7 -527.5 502.1 N2 1.106 224.0 226.0 37.3 -298.3 133.3 1.946 109.1 -813.8 742.0 O2 1.219 123.0 119.0 55.6 -226.5 83.5 1.538 61.2 -550.3 544.7 F2 1.399 42.0 36.0 54.4 -133.8 32.3 0.924 15.2 -244.5 283.6 Ne2 2.728 0.9 0.1 7.3 -3.3 0.0 0.034 -0.3 9.1 0.2 Na2 3.184 16.0 18.0 15.8 -48.1 1.1 0.758 8.6 -119.6 126.8 B2s 1.620 56.0 71.0 54.0 -161.3 26.0 1.315 28.3 -228.7 254.5 c ´ Angel Mart´ ın Pend´ as, 2005 (68)
35. IQA Chemical insights Deformation Energy: 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 2 3 4 5 6 7 8 9 10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 E/Eh Z ∆TA −∆Ven AA ∆Vee AA Ebind Edef Edef /N • Double Bell Edef curve. Half-filled shell effect. • Both ∆V AA en and ∆V AA ee have triangular shape. • Not exactly symmetric. p3 to p6 more destabilizing. • Singlet B2 has larger Edef . Minimum Edef for ground state? • Edef is one order of magnitude smaller than any of the individual E changes. • Ebind is almost exactly doubly li- near while filling the p shell. • Edef /Nel is rather constant. Ot- her ideas? c ´ Angel Mart´ ın Pend´ as, 2005 (69) IQA Chemical insights Binding Energy: −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 1 2 3 4 5 6 7 8 9 10 E/Eh Z Eint Vxc Vclas −0.50 −0.45 −0.40 −0.35 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 E/Eh fAB Eint Li Be B C N O F H Ne He 2 2 2 2 2 2 2 2 2 2 • Vxc decreases almost linearly with number of bonding electrons. • fAB is a measure of Eint . • Z = 1, 2, 7, 8, 9, 10 molecules lie on the same Eint = αfab curve. • Do different α’s depend on bonding shell, sub–shell? c ´ Angel Mart´ ın Pend´ as, 2005 (70)
36. IQA Chemical insights Correlations: 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 fAB ρb Li Be B C N O F H Ne He 2 2 2 2 2 2 2 2 2 2 0.0 20.0 40.0 60.0 80.0 100.0 0 2 4 6 8 10 12 % Eint Z Edef Eclas −0.50 −0.45 −0.40 −0.35 −0.30 −0.25 −0.20 0.32 0.36 0.40 0.44 0.48 0.04 0.08 0.12 0.16 0.20 0.24 0.02 0.04 0.06 0.08 0.10 0.12 E/Eh RAB −1 RAB −3 Eint Eclas • fAB proportional to ρb ? • The problem of F2 : Deformation • Each interaction is dominated by one multipolar term. c ´ Angel Mart´ ın Pend´ as, 2005 (71) IQA Examples 7. Other IQA examples. Ionicity, Polarity, Covalency. HF//TZV(p,d) LiH BeH2 BH3 CH4 NH3 H2 O HF Q(H) -0.9111 FAH 0.2040 FHH - EAH int -0.2886 EAH clas -0.2518 V AB X -0.0367 ∇2ρb 0.1420 -0.8684 0.2651 0.0904 -0.7009 -0.6389 -0.0620 1.4354 -0.6973 0.5143 0.1416 -0.8791 -0.7361 -0.1429 -0.3101 -0.0394 0.9835 0.0434 -0.2530 0.0347 -0.2877 -1.0595 0.3611 0.8816 0.0173 -0.4062 -0.1384 -0.2679 -1.9089 0.6035 0.6450 0.0077 -0.5551 -0.3548 -0.2002 -3.0992 0.5800 0.6762 - -0.2652 -0.1395 -0.1257 -3.8719 c ´ Angel Mart´ ın Pend´ as, 2005 (72)
37. IQA Examples Bond formation and bond breaking. H2 O CAS[6,5]//6-311G** calculation. 1.4 2.85 ¢¡ £ 1.78 ¤ ¡ ¤ a b c ¥§¦ ¨ d 1.4 1.4 a b,c,d Property a b c d QH 0.088 0.312 0.546 0.560 EOH int -0.059 -0.227 -0.518 -0.490 V OH clas 0.001 -0.052 -0.304 -0.292 V OH xc -0.060 -0.175 -0.214 -0.198 EHH int -0.138 0.016 0.252 0.124 V HH clas 0.050 0.117 0.278 0.130 V HH xc -0.188 -0.098 -0.026 -0.006 EH2 -1.018 -0.722 -0.345 -0.485 EO -74.731 -74.783 -74.577 -74.633 c ´ Angel Mart´ ın Pend´ as, 2005 (73) IQA Examples Isomers & Functional Groups: HCN vs. CNH. [ HF//TZV(p,d) ] CNH HCN 1.174 −1.749 0.575 0.022 1.742 0.664 2.168 1.857 −53.3398 −36.8493 −0.3028 −0.4175 −36.8044 −53.5601 0.212 +1.200 −1.411 0.084 0.916 2.291 1.999 2.128 Net charge F E self R(bohr) AB c ´ Angel Mart´ ın Pend´ as, 2005 (74)
38. IQA Examples Isomers & Functional Groups: HCN vs. CNH. [ HF//TZV(p,d) ] CNH HCN 1.174 −1.749 0.575 −53.3398 −36.8493 −0.3028 −0.4175 −36.8044 −53.5601 0.212 +1.200 −1.411 Net charge E self 0.2425 −2.0702 −0.5710 −0.1060 −1.8862 −0.1240 −0.2091 −0.4632 −0.5763 −0.2774 Vxc AB EAB int −0.0023 −0.0104 c ´ Angel Mart´ ın Pend´ as, 2005 (75) IQA Examples Isomers & Functional Groups: HCN vs. CNH. [ HF//TZV(p,d) ] • ∆E 10 kcal/mol. • ∆Eself (CN) < 6 kcal/mol. • Eself (CN/HCN) within 1 kcal/mol of extrapolated va- lue using the E.A. of CN (0.1420). CNH HCN 0.575 −0.3028 −0.4175 0.212 Net charge E self Vxc AB EAB int −92.2593 −0.3285 −0.2114 −0.2300 −0.2878 −92.2507 c ´ Angel Mart´ ın Pend´ as, 2005 (76)
39. IQA Examples Multiple bonds. HF//TZV(p,d) −0.039 −37.4162 −0.4381 −0.2867 +0.1455 −37.4467 −0.4220 −0.2453 −0.2924 −0.6045 −0.7544 −0.006 −37.4083 −0.4373 −0.2920 −0.4606 −0.5319 −0.2527 −0.0004 −0.056 −0.4362 −37.3846 −0.2757 −0.3021 −0.2548 −0.2858 −0.0006 0.9835 2.8678 0.9746 −0.2530 1.8950 0.9812 0.9971 0.9676 c ´ Angel Mart´ ın Pend´ as, 2005 (77) IQA Examples Multiple bonds. HF//TZV(p,d) −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 0.5 1.0 1.5 2.0 2.5 3.0 E/Eh fAB Eint VX −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.4 −0.3 −0.2 −0.1 E/Eh Eexp /Eh Estandard VX • EAB int changes by a few kcal/mol for equivalent bonds. • Interaction energies are proportional to delocalization indices. ◦ The correlation is best for V AB xc . • Standard Bond energies are proportional to Interaction Energies . ◦ The correlation is best for V AB xc . c ´ Angel Mart´ ın Pend´ as, 2005 (78)
40. IQA Examples Mixed nature bonds. COH2 , CO. HF//TZV(p,d) AB ρb ∇2ρb Q(A) Q(B) rb (A) rb (B) CO (COH2 ) 0.4497 0.0671 1.324 -1.263 0.743 1.483 CO (CO) 0.5400 0.5490 1.353 -1.353 0.705 1.380 CH (COH2 ) 0.2984 -1.1723 1.324 -0.032 0.726 1.302 c ´ Angel Mart´ ın Pend´ as, 2005 (79) IQA Examples Mixed nature bonds. COH2 , CO. HF//TZV(p,d) Net charge F E self R(bohr) AB −1.263 +1.324 −0.032 1.376 0.899 0.052 −0.4367 −0.2362/−0.2751 −0.0051/−0.0059 −74.2257 −36.7540 +1.353 −1.353 1.572 −36.7116 −74.0718 −2.0078/−0.4402 −1.5356/−0.4055 E / V int X 2.085 2.226 2.063 c ´ Angel Mart´ ın Pend´ as, 2005 (80)
41. IQA Summary & Conclusions 8. Summary, Conclusions, Perspectives. Most Quantitative theories of the chemical bond rely on a fragment picture. They usually need external references. We may get rid of the reference by means of intrinsic partitions. Among these, a partition of the molecular density into atomic contributions is a desirable goal: Atoms in Molecules • Once obtained, (in absence of the exact Density Functional) we need a partition of ρ2 . • Li & Parr proposed a consistent way to do that: Do not discriminate electrons depending on the atoms to which they are ascribed. • This recipe provides an energy partition for any density partition. • Among many possibilities, QTAM atoms minimize their additive energies while preserving QM. c ´ Angel Mart´ ın Pend´ as, 2005 (81) IQA Summary & Conclusions The IQA approach is chemicaly intuitive. • Atoms do maintain their individuality. • There is Charge Transfer and Promotion (deformation) plus interaction. • All Quantum atoms interact with each other. • All the energy terms have clear physical meaning. • Exchange maintains its original physical meaning, and is not mixed up with orthogonality requirements. • Interaction is the sum of classical (QM–compatible) and Quantum contributions. • Vclas is positive for homopolar interactions. May become negative if there is CT. • Homopolar interactions rest in QM, non–classical, exchange–correlation forces. • A classification of interactions compatible with our background is obtained. • Functional groups may be recognized as hardly changing Eself groups of atoms. • Interaction energies correlate with standard bond energies. • The approach is general. As much correlation as desired may be introduced in Ψ. • ... and compatible with QTAM: bonded atoms correspond to preferred interaction channels. c ´ Angel Mart´ ın Pend´ as, 2005 (82)
42. IQA Summary & Conclusions The IQA approach is NOT the solution. It is another TOOL at hand: • It is not cheap: Computationally expensive. Big molecules out of present implemen- tations. • At the moment, integration accuracy is sometimes hard to achieve. • It lies out of DFT. Perhaps with future implementations of density matrix functionals? • You have to be comfortable with QTAM. Some people are not. Then perhaps try your favorite partition method? • It provides magnitudes invariant under orbital transformations. You don’t have a partition into orbital contributions. Perspectives: • Use spin(full) ρ2 . • Use other thinner basins: ELF, ... • Build libraries of energies, chemical potentials, multipoles, and use them as molecular building blocks. • Examine systems. c ´ Angel Mart´ ın Pend´ as, 2005 (83) IQA Summary & Conclusions Bibliography: R. F. W. Bader, Atoms in Molecules. Oxford Univ. Press. Oxford 1990. L. Li and R. G. Parr, J. Chem. Phys. 84, 1704 (1986). J. Rychlewski and R. G. Parr, J. Chem. Phys. 84, 1696 (1986). K. Ruedenberg, Rev. Mod. Phys., 34, 326 (1962). F. M. Bickelhaupt and E. J. Baerends, Rev. Comput. Chem., 15, 1 (2000). G. Frenking et. al., Coord. Chem. Rev. 238-239, 55 (2003). S. Shaik et. al., J. Am. Chem. Soc. 114, 7861 (1992). A. Mart´ ın Pend´ as, Aurora Costales and V´ ıctor Lua˜ na. Phys. Rev. B 55, 4275 (1997). A. Mart´ ın Pend´ as, M. A. Blanco, and E. Francisco. J. Chem. Phys. 120, 4581 (2004). A. Mart´ ın Pend´ as, E. Francisco, and M. A. Blanco. J. Comput. Chem. 26, 344 (2005). M. A. Blanco, A. Mart´ ın Pend´ as, and E. Francisco, J. Chem. Phys., submitted. c ´ Angel Mart´ ın Pend´ as, 2005 (84)