Response to “Comment on ‘Correlated electron-nuclear dynamics: Exact
factorization of the molecular wavefunction”' [J. Chem. Phys. 139, 087101
(2013)]
Ali Abedi, Neepa T. Maitra, and E. K. U. Gross
Citation: J. Chem. Phys. 139, 087102 (2013); doi: 10.1063/1.4818523
View online: http://dx.doi.org/10.1063/1.4818523
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THE JOURNAL OF CHEMICAL PHYSICS 139, 087102 (2013)
Response to “Comment on ‘Correlated electron-nuclear dynamics:
Exact factorization of the molecular wavefunction”’ [J. Chem. Phys. 139,
087101 (2013)]
Ali Abedi,1,2 Neepa T. Maitra,3 and E. K. U. Gross1,2
1
Max-Planck Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
European Theoretical Spectroscopy Facility (ETSF)
3
Department of Physics and Astronomy, Hunter College and the City University of New York, 695 Park Avenue,
New York, New York 10065, USA
2
(Received 30 July 2013; accepted 1 August 2013; published online 28 August 2013)
[http://dx.doi.org/10.1063/1.4818523]
Reference 1 raises the question of whether the equations for the electronic and nuclear wavefunctions derived in
Ref. 2 preserve the norms of these wavefunctions, and provides a proof that the equations do in fact preserve the norms.
There is another way to see that the norms are preserved by
the time evolution, which was indicated in the original earlier work given in Ref. 3, although it was not fleshed out in
Ref. 2. In this earlier work where the exact factorization was
first presented, the derivation of the equations is outlined in
the following statement: “We require the action to be stationary with respect to variations in R (r, t) and χ (R, t) subject
to the condition (5).” where condition (5) is the partial normalization condition (PNC) of Ref. 3, and the action is the
t
Frenkel action S[, ∗ ] = ti f dt|Ĥ − i∂t |, with Ĥ the
Hamiltonian of the full molecular system. A common practise in variational methods is to enforce such conditions via
the method of Lagrange multipliers. Reference 2 provides the
details of the stationarizing procedure, but the derivation appears to neglect the term that enforces the normalization constraint. In fact, the Lagrange multiplier turns out to be zero as
we shall show below explicitly.
One adds a term with Lagrange multiplier μ(Rσ , t) to the
action of Eq. (37) in Ref. 2, to enforce the PNC, defining
S̃[Rσ , ∗Rσ , χ , χ ∗ ]
= S[Rσ , ∗Rσ , χ , χ ∗ ]
+
dRμ(Rσ , t)
dr|Rσ (rs, t)|2 , (1)
s
σ
and performs the variations of S̃[Rσ , ∗Rσ , χ , χ ∗ ] with respect to the electronic and nuclear wavefunctions. Requiring
0 = δ S̃/δχ ∗ yields
1
− i∂t |Rσ χ +
(−i∇α χ /χ ) · Aα χ
Mα
α
−∇ 2
α
n
+ V̂ext − i∂t χ
= −Rσ |Rσ (2)
2Mα
α
Rσ |Ĥenew
0021-9606/2013/139(8)/087102/2/$30.00
while requiring 0 = δ S̃/δ∗Rσ yields
new
Ĥe − i∂t + μ/|χ |2 Rσ
1
(−i∇α χ /χ ) · (−i∇α Rσ )
Mα
α
−∇α2
n
+
V̂
−
i∂
t χ
ext
α 2Mα
· Rσ ,
=−
(3)
χ
−∇ 2
e
where Ĥenew ≡ ĤBO + V̂ext
+ α 2Mαα and the notation follows that of Ref. 2. Replacing the right-hand side of Eq. (3) by
the left-hand side of Eq. (2), multiplied by Rσ /(Rσ |Rσ ·
χ ) we arrive at
new
Ĥe − i∂t +μ/|χ |2 Rσ
+
+
=
1
(−i∇α χ /χ ) · (−i∇α Rσ )
Mα
α
Rσ |Ĥenew −i∂t |Rσ + α
1
(−i∇α χ/χ )
Mα
· Aα Rσ
Rσ |Rσ .
(4)
At first sight the procedure appears to add a term
μ(Rσ , t) to the resulting electronic Hamiltonian. However,
multiplying Eq. (4) by ∗Rσ (rs, t) and integrating over rs,
readily determines the value of the Lagrange multiplier:
μ(Rσ , t) = 0.
Equation (4) was derived using the method of Lagrange
Multipliers to enforce the PNC: it therefore preserves the
norm, and so starting with a state Rσ (0) that has unit norm
Rσ (0)|Rσ (0)rs = 1 means that it will evolve with unit
norm Rσ (t)|Rσ (t)rs = 1. Since μ(Rσ , t) = 0, Eq. (4)
then reduces to the electronic equation (28) of Ref. 2. There
is also no change in the nuclear equation. Hence, performing
the variation of the action under the partial normalization constraint yields no change to any of the equations or statements
in Ref. 2. Including or not including the partial normalization condition as Lagrangian constraint in the variation of the
action functional makes no difference in the resulting equations of motion because the Lagrangian multiplier μ(Rσ , t)
vanishes identically.
139, 087102-1
© 2013 AIP Publishing LLC
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087102-2
Abedi, Maitra, and Gross
Partial support from the Deutsche Forschungsgemeinschaft (SFB 762), the European Commission
(FP7-NMP-CRONOS), and from the National Science Foundation (CHE-1152784) (N.T.M.) is gratefully
acknowledged.
J. Chem. Phys. 139, 087102 (2013)
1 J.
L. Alonso, J. Clemente-Gallardo, P. Echenique-Robba, and J. A. JoverGaltier, J. Chem. Phys. 139, 087101 (2013).
2 A. Abedi, N. T. Maitra, and E. K. U. Gross, J. Chem. Phys. 137, 22A530
(2012).
3 A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002
(2010).
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