Molecular structure calculationswithout the Born–Oppenheimer approximation

Edit Mátyus1,2 and Markus Reiher2

1 Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, Eötvös University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary

2 Laboratory of Physical Chemistry, ETH Zürich, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland

E-mail: matyus@chem.elte.hu, markus.reiher@phys.chem.ethz.ch

Introduction

Background

•Most of the present-day theoretical and computational chemistry is basedon the clamped nuclei or Born–Oppenheimer (BO) paradigm.

• This work is devoted to a theoretical approach which does not rely on thisparadigm.

Questions

• Can we reconstruct fundamental chemical concepts in a non-BO theory?

• Can we calculate accurate energy levels and wave functions using thepresent-day computational resources for the ground and excited states ofmolecules with various quantum numbers of the non-relativistic theory?

Starting point

• Our goal here is the development of a variational procedure for the accuratesolution of the Schrödinger equation of few-particle systems.

•We describe electrons and atomic nuclei (or other particles) on equal footing,

Non-Born–Oppenheimer (NBO) theory

Non-relativistic quantum mechanics for n+1 particles

• Schrödinger equation with parameters mi and qi

H = −n+1∑

i=1

12mi

∆r i +n+1∑

i=1

n+1∑

j>i

qiqj

|r i − r j |

• Pauli principle for fermions and bosons with spins si

⇒ 3(n+1) physical parameters: mass, electric charge, and spin

Invariance properties and quantum numbers

• Translational invariance⇒ Translationally invariant coordinates or translationally invariant parame-terization

• Rotation-inversion invariance⇒ Rotational or orbital angular momentum, L,ML, and parity, p

• Permutation invariance of identical particles+ Pauli principle⇒ spin angular momentum S,MS

NBO algorithm: a multi-stage variational pro-cedure

1. Coordinates

• Translationally invariant or laboratory-Vxed Cartesian coordinates, r

2. Quantum Hamiltonian

• Expression in terms of (transl. inv., rectilinear) Cartesian coordinates

3. “Well-parameterized”, symmetry-adapted basis functions

ΦLMLpSMS(r ,σ; A, u, K ,θ) = A{φLMLp(r ; A, u, K ) χSMS

(σ;θ)}

with A = N−1/2p

∑Npi=1 εiPi.

• Eigenfunctions of L2, Lz, p, S2, and Sz

• Gaussian geminals with polynomial prefactors

• Global vector representation for the angular motion

φLMLp(r ;α, u, K ) = |v |2K +LYLML(v ) exp

−12

n+1∑

i=1

n+1∑

j>i

αij(r i − r j)2

, for p = (−1)L

with v =∑n

i=1 uir i ∈ R3, v = v/|v |, K ∈ N0.

4. Overlap and Hamiltonian matrix

• Analytic expressions

• Numerically stable implementation for large values of the polynomial ex-ponents (2K + L > 4)

5. Eigensolver

• Linear combination of (n + 1)-particle basis functions

ΨLMLpSMS(r ,σ; c,α, u, K ,θ) =

Nb∑

I=1

cI ΦLMLpSMS(r ,σ;αI, uI, KI,θI)

• Generalized eigensolver (Intel MKL) for < 1 500×1 500 symmetric matrices

• The number of basis functions, Nb, is increased until convergence

5+1. On-the-Wy optimization of the basis function parameters

• Non-linear parameters, αI, uI, KI, ..., are optimized when Vrst generated andreoptimized regularly during the calculation

• Stochastic variational procedure with a sampling-importance-resamplingstrategy

NBO computer program

A computer program was developed using the FORTRAN language

• Input parameters: mi , qi , si (i = 1, 2, ... , n + 1)

• Eigenstates corresponding to diUerent quantum numbers are obtained inseparate runs (L = 0, 1, 2, ..., ML = −L, ... , +L, p = +1,−1, and S, MS)

• Physical properties are calculated using the wave function

Molecular structure

Classical concepts of molecular structure vs. holistic nature of quantummechanics

• recognize elements of the classical molecular structure using few-particledensity functions

“1-(pseudo)particle” density⇒ radial density

D(1)P,a(R1) = 〈Ψ(r)|δ(ra − rP − R1)|Ψ(r)〉

ρP,a(R) = D(1)P,ab(R1) with R1 = (0, 0, R)

×R

“2-(pseudo)particle” density⇒ angular density

D(2)P,ab(R1, R2) = 〈Ψ(r)|δ(ra − rP − R1)δ(rb − rP − R2)|Ψ(r )〉

ΓP,ab(α) =∫ ∞

0dR1R2

1

∫ ∞

0dR2R2

2 D(2)P,ab(R1, R2) with RT

1R2 = R1R2 cosα

× α

On the emergence of molecular structure

Mass-scale similarity of the Coulomb Hamiltonian

Scaling the masses in the non-relativistic Coulomb Hamiltonian

H(ma, mb, ... , r) → H(ηma, ηmb, ... , r) = ηH(ma, mb, ... , ηr)

is equivalent to scaling the energy and the length

E → ηE and ψ(r) → ψ(ηr)

Radial density vs. mass ratio (ma/mb) in {a±, a±, b∓}-type systems (g.s.,L = 0)

mamb

: 0.000 543 0.5 1 2 5 1840

H− Ps− H+2

{e−, e−, p+} {e−, e−, e+} {p+, p+, e−}

⇒ atomic vs. molecular structure ∼ mass-polarization eUects

• transition point: 0.5 < ma/mb < 1.0

• “alchemical transformation” with respect to the mass ratio

• Ps−: molecular-type system

• H+2: “zero-point vibrations”, ball-like shape for L = 0

• classical vs. quantum structure?

EM, J. Hutter, U. Müller-Herold, and M. Reiher,

Phys. Rev. A 83, 052512 (2011); J. Chem. Phys. 135, 204302 (2011).

Extracting elements of molecular structurefrom the all-particle wave function

Radial and angular densities vs. mass ratio (ma/mb)

in {a±, a±, b∓}- and {a±, a±, b∓, b∓}-type systems (g.s., L = 0)

{e−, e−, p+} {e−, e−, e+} {p+, p+, e−} {e−, e−, p+, p+} {e−, e−, e+, e+} {p+, p+, e−, e−}

ma/mb 0.000 543 1 1 840 0.000 543 1 1 840

ρ0,a

ρ0,b

Γ0,aa

Γ0,bb

⇒ molecular structure ∼ strong correlation eUects for the nuclei

Radial and angular density, H2D+ = {e−, e−, p+, p+, d+} (g.s., L = 0)

ρ0,e ρ0,p ρ0,d ρd,p

Γ0,ee′ Γ0,pp′ Γ0,dp Γd,pp′

SutcliUe and Woolley; CaVero and Adamowicz:

or〈α〉 = (0o+ 180o+ 0o)/3 = 60o 〈α′〉 = (60o+ 60o+ 60o)/3 = 60o

EM, J. Hutter, U. Müller-Herold, and M. Reiher,

Phys. Rev. A 83, 052512 (2011); J. Chem. Phys. 135, 204302 (2011).

Rotation-vibration-electronic excited states

H+2 = {p+, p+, e−}

−0.60

−0.59

−0.58

−0.57

−0.60

−0.59

−0.58

−0.57

−0.49975

−0.49970

−0.49975

−0.49970

Energy[E

h]

A 2Σ+u

−0.49975

−0.49970

Energy[E

h]

A 2Σ+u

X 2Σ+g

(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)

(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)

X 2Σ+g

(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)

(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)

(0, 1,1)(1,-1,0)(2, 1,1)

(L, p, Sp)H (n = 1)

(0, 1,1)(1,-1,0)(2, 1,1)

(L, p, Sp)H (n = 1)

x

x

↑

ortho

even L: A 2Σ

+u

odd L: X 2Σ

+g

x

y

↑

para

even L: X 2Σ

+g

odd L: A 2Σ

+u

Calculated energy levels and dissociation energies of H+2 (examples)

L p Sp E / Eh D / cm−1 Assignment

2 1 1 −0.499 731 516(7) 0.807(1) A 2Σ

+u , v = 0; ortho

...0 1 0 −0.597 139 060(4) 21 379.290(2) X 2

Σ+g , v = 0; para

Refs. Karr and Hilico (2006); Moss (1993)

Ps2 = {e+, e+, e−, e−}

Calculated energy levels of Ps2

L p S+ S− E / Eh η δE / µEh Ref. Assignment

0 1 0 0 −0.516 003 788 7 3.8 · 10−9 −0.001 7 [1] 0+ A1

0 1 1 0 −0.330 287 496 4 3.3 · 10−8 +10.686 [2] 0+ E1 −1 0 0 −0.334 408 295 3 4.8 · 10−8 −0.000 2 [3] 1− B2

Refs. [1] Bubin and Adamowicz (2006); [2] Suzuki and Usukura (2000); [3] Bubin andAdamowicz (2008)

H2 = {p+, p+, e−, e−}

-1.18

-1.08

-0.98

-1.18

-1.08

-0.98

-1.18

-1.08

-0.98

-0.75

-0.65

-0.55

-0.75

-0.65

-0.55

-0.75

-0.65

-0.55

Energy

[Eh] -0.75

-0.65

-0.55

Energy

[Eh]

X 1Σ+g

b 3Σ+u H(n = 1)+H(n = 1)

X 1Σ+g

b 3Σ+u H(n = 1)+H(n = 1)

(0, 1,0,0)(1,-1,1,0)(2, 1,0,0)(3,-1,1,0)(4, 1,0,0)

(0, 1,1,1), (1,-1,0,1)

B 1Σ+u

a 3Σ+g

H(n = 1)+H(n = 2)

(0, 1,1,0)(1,-1,0,0)

(0, 1,0,1)(1,-1,1,1)

(L, p, Sp, Se)

B 1Σ+u

a 3Σ+g

H(n = 1)+H(n = 2)

(0, 1,1,0)(1,-1,0,0)

(0, 1,0,1)(1,-1,1,1)

(L, p, Sp, Se)

x

y

↑

↓singlet, para

even L: X 1Σ

+g

odd L: B 1Σ

+u

x

x

↑

↑triplet, ortho

even L: b 3Σ

+u

odd L: a 3Σ

+g

x

x

↑

↓singlet, ortho

even L: B 1Σ

+u

odd L: X 1Σ

+g

x

y

↑

↑triplet, para

even L: a 3Σ

+g

odd L: b 3Σ

+u

Calculated energy levels of H2

L p Sp Se E / Eh η δE / µEh Ref. Assignment

0 1 0 0 −1.164 025 026 1.4 · 10−8 −0.004 [4] para singlet X 1Σ

+g

1 −1 1 0 −1.163 485 167 3.2 · 10−9 −0.006 [5] ortho singlet X 1Σ

+g

2 1 0 0 −1.162 410 402 2.2 · 10−8 −0.007 [5] para singlet X 1Σ

+g

3 −1 1 0 −1.160 810 486 7.9 · 10−9 −0.006 [5] ortho singlet X 1Σ

+g

4 1 0 0 −1.158 699 660 6.7 · 10−9 −0.006 [5] para singlet X 1Σ

+g

[...]

0 1 1 1 [−0.999 449] [3.9 · 10−5] [−6.6] [6] ortho triplet b 3Σ

+u

1 −1 0 1 [−0.999 444] [3.5 · 10−5] [−11.7] [6] para triplet b 3Σ

+u

[...]

0 1 1 0 −0.753 026 938 1.8 · 10−6 −0.455 [7] ortho singlet B 1Σ

+u

1 −1 0 0 −0.752 848 338 5.9 · 10−6 −2.041 [7] para singlet B 1Σ

+u

[...]

0 1 0 1 −0.730 825 001 1.2 · 10−6 −0.196 [8] para triplet a 3Σ

+g

1 −1 1 1 −0.730 521 131 8.2 · 10−7 −0.279 [8] ortho triplet a 3Σ

+g

[...]

Refs. [4] Bubin and Adamowicz (2003); [5] Pachucki and Komasa (2009); [6] Kołos and Rych-lewski (1990); [7] Wolniewicz, Orlikowski, and Staszewska (2006); [8] Wolniewicz (2007)

7Li = {7Li3+, e−, e−, e−}

Calculated energy levels of 7Li

L p Se E η δE / µEh Ref.

0 1 1/2 −7.477 451 901 8.5 · 10−10 −0.029 [9]1 −1 1/2 −7.409 557 348 9.4 · 10−9

2 1 1/2 −7.334 926 959 7.5 · 10−9 −0.347 [10]

Refs. [9] Stanke, Kedziera, Bubin, and Adamowicz (2007);[10] Sharkley, Bubin, and Adamowicz (2011)

EM and M. Reiher, J. Chem. Phys. to be submitted (2012).

Acknowledgement

E.M. acknowledges funding by a two-year ETH Fellowship during 2010–2011.Financial support from the Swiss National Fund (SNF) is also gratefully ac-knowledged. E.M. also thanks the Hungarian ScientiVc Research Fund (OTKA,NK83583) and the European Union and the European Social Fund (TÁMOP-4.2.1/B-09/1/KMR-2010-0003) for Vnancial support during 2012.