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Anion Electronic Structure and Correlated, One-electron Theory. J. V. Ortiz Department of Chemistry and Biochemistry Auburn University Workshop on Molecular Anions and Electron-Molecule Interactions in Honor of Professor Kenneth Jordan July 1, 2007

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Anion Electronic Structure and Correlated, One-electron Theory

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Anion electronic structure and correlated one electron theory

Anion Electronic Structure and Correlated, One-electron Theory

J. V. Ortiz

Department of Chemistry and Biochemistry

Auburn University

Workshop on Molecular Anions and Electron-Molecule Interactions in Honor of

Professor Kenneth Jordan

July 1, 2007

Park City, Utah



National Science Foundation

Defense Threat Reduction Agency


  • Symposium Organizers

  • Jack Simons

  • Brad Hoffman

  • Auburn University

  • Department of Chemistry and Biochemistry

  • Auburn Coworkers

  • UNAM Collaborators:

  • Ana Martínez

  • Alfredo Guevara

Quantum chemistry s missions

Deductive agenda:

Deduce properties of molecules from quantum mechanics

Calculate chemical data, especially if experiments are difficult or expensive

Inductive agenda:

Identify and explain patterns in structure, spectra, energetics, reactivity

Deepen and generalize the principles of chemical bonding

Quantum Chemistry’s Missions

G. N. Lewis

E. Schrödinger

Anion electronic structure and correlated one electron theory

Electron Propagator




Molecular Orbital



One electron equations

Hartree Fock Theory

Hartree Fock Equations:

(Tkin + Unucl + JCoul - Kexch)φiHF ≡

F φiHF=εiHF φiHF

Same potential for all i:

core, valence, occupied, virtual.

εiHF includes Coulomb and exchange contributions to IEs and EAs

Electron Propagator Theory

Dyson Equation:

[F + ∑(εiDyson)]φiDyson = εiDyson φiDyson

Self energy, ∑(E): Energy dependent, nonlocal potential that varies for each electron binding energy

εiDyson includes Coulomb, exchange, relaxation and correlation contributions to IEs and EAs

φiDyson describes effect of electron detachment or attachment on electronic structure

One-electron Equations

Dyson orbitals feynman dyson amplitudes

Dyson Orbitals (Feynman-Dyson Amplitudes)

  • Electron Detachment (IEs)

    φiDyson(x1) =



  • Electron Attachment (EAs)

    φiDyson(x1) =

    (N+1)-½∫ Ψi,N+1(x1,x2,x3,...,xN+1)Ψ*N(x2,x3,x4,…,xN+1) dx2dx3dx4…dxN+1

  • Pole strength

    Pi = ∫|φiDyson(x)|2dx

    0 ≤ Pi ≤ 1

Electron propagator concepts

Electron Propagator Concepts

Electron Correlation

Dyson Orbital

Canonical MO

Correlated Electron

Binding Energy

Orbital Energy

Integer Occupation


Pole Strengths





Accuracy versus interpretability

Accuracy versus Interpretability

  • Does electron propagator theory offer a solution to Mulliken’s dilemma?

The more accurate the

calculations become,

the more the concepts

vanish into thin air.

- R. S. Mulliken

Substituent effects u and t

Substituent Effects: U and T

Dyson orbitals for u and t ies

Dyson Orbitals for U and T IEs








Methyl (CH3) participation

Uracil versus thymine

Uracil versus Thymine

  • Methyl group destabilizes π orbitals with large amplitudes at nearest ring atom

  • Therefore, IE(T) < IE(U)

  • Valid principles for substituted DNA bases, porphyrins and other organic molecules

A self energy for large molecules p3

A Self-Energy for Large Molecules: P3

  • Neglect off-diagonal elements of Σ(E) in canonical MO basis: φiDyson(x) = Pi½φiHF-CMO(x)

  • Partial summation of third-order diagrams

  • Arithmetic bottleneck: oN4 (MP2 partial integral transformation)

  • Storage bottleneck: o2v2 in semidirect mode

  • Abelian, symmetry-adapted algorithm in G03

Formulae for p3 e

Formulae for ΣP3(E)

ΣP3pq(E) =

½Σiab <pi||ab><ab||qi> Δ(E)-1iab +

½Σaij <pa||ij>(<ij||qa> + Wijqa) Δ(E)-1aij +

½Σaij Upaij(E)<ij||qa>Δ(E)-1aij


Δ(E)-1pqr = (E + εp – εq – εr)-1

Wijqa = ½Σbc<bc||qa><ij||bc> Δ-1ijbc

+ (1-Pij)Σbk<bi||qk><jk||ba> Δ-1jkab

Upaij(E) = - ½Σkl<pa||kl><kl||ij> Δ(E)-1akl

- (1 – Pij) Σbk<pb||jk><ak||bi> Δ(E)-1bjk

P3 performance

P3 Performance

  • 31 Valence IEs of Closed-Shell Molecules:


    MAD (eV) = 0.20 (tz)

  • 10 VEDEs of Closed-Shell Anions:


    MAD (eV) = 0.25 (a-tz)

  • Arithmetic bottleneck: o2v3 for Wijqa

  • Storage bottleneck: <ia||bc> for Wijqa

Recent applications porphyrins and fullerenes

Recent Applications: Porphyrins and Fullerenes

Invitation to propagate

Invitation to Propagate

Input to Gaussian 03

# OVGF 6-311G** iop(9/11=10000)

P3 Electron Propagator for Water

0 1


H 1 0.98

H 1 0.98 2 105.

Available diagonal approximations for Σ(E):

Second order, Third order, P3, OVGF (versions A, B & C)

Nucleotides gaseous spectra

Nucleotides: Gaseous Spectra

  • Nucleotides: phosphate-sugar-base DNA fragments

  • Electrospray ion sources

  • Magnetic bottle detection

  • High resolution laser spectroscopy of ions, mass spectrometry

  • Goal: predict photoelectron spectra of anionic nucleotides (vertical electron detachment energies or VEDEs)

Photoelectron spectra of 2 deoxybase 5 monophosphate anions

Photoelectron Spectra of 2’-deoxybase 5’-monophosphate Anions


Anomalous peak for dGMP

Base = adenine


G: lowest IE

of DNA bases

Base = cytosine


Base = guanine

Dyson orbitals for

lowest VEDEs:

phosphate or base?


Base = thymine

L-S.Wang, 2004

Damp isomers and energies

DAMP Isomers and Energies

0 kcal/mol



Damp vedes ev and dyson orbitals

DAMP VEDEs (eV) and Dyson Orbitals

Dgmp isomers and energies

DGMP Isomers and Energies

0 kcal/mol



Dgmp vedes ev and dyson orbitals

DGMP VEDEs (eV) and Dyson Orbitals

Hydrogen bonds dgmp vs damp

Hydrogen Bonds: DGMP vs DAMP

  • DGMP: G amino to Phosphate oxygen

  • DAMP: Sugar hydroxy to Phosphate oxygen

Nucleotide electronic structure

Nucleotide Electronic Structure

  • Phosphate anion reduces Base VEDEs by several eV

  • Base also increases Phosphate VEDEs

  • Therefore, Base and Phosphate VEDEs

    are close

  • Differential correlation effects are large

  • Koopmans ordering is not reliable

A simple renormalized self energy p3

A Simple, Renormalized Self-Energy: P3+

ΣP3+pq(E) =

½Σiab <pi||ab><ab||qi> Δ(E)-1iab +

[1+Y(E)]-1 ½Σaij<pa||ij>(<ij||qa> + Wijqa) Δ(E)-1aij + ½Σaij Upaij(E)<ij||qa>Δ(E)-1aij


Y(E) = {-½Σaij<pa||ij>Wijqa Δ(E)-1aij} {½Σaij<pa||ij><ij||qa> Δ(E)-1aij}-1

P3 performance1

P3+ Performance

  • 31 Valence IEs of Closed-Shell Molecules:


    MAD (eV) = 0.19 (tz), 0.19 (qz)

  • 10 VEDEs of Closed-Shell Anions:


    MAD (eV) = 0.11 (a-tz), 0.13 (a-qz)

Reactivity of al 3 o 3 with h 2 o

Reactivity of Al3O3- with H2O

  • Wang: first anion photoisomerization

  • Jarrold: Al3O3-(H2O)n photoelectron spectra n=0,1,2

  • Distinct profile for n=1

  • Similar spectra for n=2 and n=0

Al 3 o 3 photoelectron spectrum

Al3O3- Photoelectron Spectrum



Cluster vedes and dyson orbitals

Cluster VEDEs and Dyson Orbitals




Strong initial state correlation

Strong Initial State Correlation

  • Need better reference orbitals for:

    diradicaloids, bond dissociation, unusual bonding …

  • Generate renormalized self-energy with approximate Brueckner reference determinant

A versatile self energy bd t1

A Versatile Self-Energy: BD-T1

  • Asymmetric Metric:



  • Galitskii-Migdal energy =

    BD (Brueckner Doubles, Coupled-Cluster)

  • Operator manifold: f~a†aa=f3

  • Discard only 2ph-2hp couplings

Applications of the bd t1 approximation

Applications of theBD-T1 Approximation

  • Vertical Electron Detachment Energies of Anions: MAD=0.03 eV

  • 1s Core Ionization Energies: MAD = 0.2%

  • Valence IEs of Closed-Shell Molecules:

    MAD = 0.15 eV

  • IEs of Biradicaloids: MAD = 0.08 eV

Bowen s photoelectron spectrum of nh 4

Bowen’s Photoelectron Spectrum of NH4-

B: Mysterious low-VEDE peak

Not due to hot NH4-

Variable relative intensity

Another isomer of NH4-?

A: H- detachment

with vibrational

excitation of NH3

X: H-(NH3)

NH3 increases H- VEDE





Computational search nh 4 structures

Computational Search: NH4- Structures

Hydride anion: H-

H-(NH3) constituents:

Ammonia molecule: NH3

Lewis: 1 electron pair

H nucleus has 1+ charge

Negative charge attracts

+ end of polar NH bond

Lewis: 3 electron pairs

shared in polar NH bonds

+ 1 unshared pair on N

Partial + charge on H’s

Partial – charge on N



accounts for

dominant peaks

Computational search what is the structure for the low vede peak

Computational Search:What is the structure for the low-VEDE peak?

Idea: NH2-(H2) anion-molecule complex

Reject: spectral peak would be high-VEDE, not low

Idea: NH4- has 5 valence e- pairs

Deploy in 4 N-H bonds and 1 unshared pair

at the 5 vertices of a trigonal biprism or

square pyramid

Calculations find no such structures!

Instead, they spontaneously rearrange ….

To a heretical structure

….to a heretical structure!

Tetrahedral NH4- has 4

equivalent N-H bonds

Defies Lewis theory

Defies valence shell

electron pair

repulsion theory

Structure similar to that of NH4+

So where are the 2 extra electrons?

Structural confirmation experiment and theory

Structural Confirmation:Experiment and Theory

Predicted VEDEs from Electron Propagator Theory

for Anion(molecule) and Tetrahedral forms of NH4-

coincide with peaks from photoelectron spectrum

Dyson orbitals for vedes of nh 4

Dyson Orbitals for VEDEs of NH4-

H-(NH3) has 2 electrons

in hydride-centered orbital

with minor N-H delocalization.

VEDE is 1.07 eV

Tetrahedral NH4- has 2

diffuse electrons located

chiefly outside of NH4+ core.

VEDE is 0.47 eV

Irc t d nh 4 h nh 3

IRC: Td NH4- -> H-(NH3)

Energy (au)

Intrinsic Reaction Coordinate

Double rydberg anions

Double Rydberg Anions

  • Highly correlated motion of two diffuse (Rydberg) electrons in the field of a positive ion (NH4+ , OH3+)

  • United atom limit is an alkali anion: Na-

  • Extravalence atomic contributions in Dyson orbitals



Irc c 3v oh 3 h h 2 o

Eact = 5.1

Erx = -39.9

IRC: C3v OH3- -> H-(H2O)

Bowen s photoelectron spectrum of n 2 h 7

Bowen’s Photoelectron Spectrum of N2H7-

X: H-(NH3)2 e- detachment

B & C: two low EBEs!






Calculated n 2 h 7 structures

Calculated N2H7- Structures

  • H-(NH3)2 anion-

    molecule complex

  • NH4-(NH3) anion-

    molecule complex

    with tetrahedral NH4-

  • N2H7- with hydrogen bond (similar to N2H7+ )

N 2 h 7 vedes and dyson orbitals

N2H7- VEDEs and Dyson Orbitals

H-(NH3)2 has hydride centered Dyson orbital

EPT predicts 1.49 eV for VEDE

Peak observed in spectrum at 1.46 ± 0.02 eV

Dyson orbital concentrated near NH4-

EPT predicts 0.60 eV for VEDE

Peak observed at 0.58 ± 0.02 eV

Dyson orbital concentrated near 3 hydrogens

EPT predicts 0.42 eV for VEDE

Peak observed at 0.42 ± 0.02 eV

Assignment of n 3 h 10 ebes to double rydberg anions

Assignment of N3H10-EBEs to Double Rydberg Anions

  • (NH4-)(NH3)2 : 0.66 (Expt.) 0.68 (EPT)

  • (N2H7-)(NH3) : 0.49 (Expt.) 0.49 (EPT)

  • (N3H10-) : 0.42 (Expt.) 0.40 (EPT)


O 2 h 5 and n 2 h 7 structures

O2H5- and N2H7- Structures




O 2 h 5 vedes and dyson orbitals

O2H5- VEDEs and Dyson Orbitals

H-(H2O)2 VEDE: 2.36 eV

H-bridged VEDE: 0.48 eV

Ion-dipole VEDE: 0.74 eV

Electron pair concepts old and new

Electron Pair Concepts: Old and New

Chemical bonds arise

from pairs of electrons

shared betweenatoms

G.N. Lewis

I. Langmuir

W.N. Lipscomb

Unshared pairs

localized on single atoms

affect bond angles

Molecular cations may

bind an e- pair peripheral

to nuclear framework:

Double Rydberg Anions

R.J. Gillespie

R.S. Nyholm

Electron propagator theory and quantum chemistry s missions

Electron Propagator Theory and Quantum Chemistry’s Missions

  • Deductive, quantitative theory:

    Prediction and interpretation enable dialogue with experimentalists requiring accurate data

  • Inductive, qualitative theory:

    Orbital formalism generalizes and deepens qualitative notions of electronic structure, relating structure, spectra and reactivity

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