Reply to Comment on the Paper ‘‘An Efficient Algorithmfor Energy Gradients and Orbital Optimization inValence Bond Theory’’

Wei Wu*[a] and Yirong Mo[b]

van Lenthe, Broer, and Rashid made comments on our 2009

paper [Song et al., J. Comput. Chem. 2009, 30, 399] by criticizing

that we did not properly reference the work by Broer and

Nieuwpoort in 1988 [Broer and Nieuwpoort, Theor. Chim. Acta.

1988, 73, 405], and we favorably compared our valence bond

self-consistent field (VBSCF) algorithm with theirs. However, both

criticisms are unjustified insignificant. The Broer–Nieuwpoort

algorithm, properly cited in our paper, is for the evaluations of

matrix elements between determinants of nonorthogonal

orbitals. Stating that this algorithm ‘‘can be used for an orbital

optimization’’ afterwards [van Lenthe et al., submitted] is not a

plausible way to require more credits or even criticize others.

While we stand by our statement that our algorithms scales at

O(m4) and van Lenthe et al.’s approximate Newton Raphson

algorithm scales at O(mN5) (here m and N are the numbers of

basis functions and electrons), as we discussed in our original

paper, it becomes obvious that any strict comparison among

different algorithms is difficult, unproductive, and counteractive.VC 2012 Wiley Periodicals, Inc.

DOI: 10.1002/jcc.22923

The Comment by van Lenthe, Broer, and Rashid [J. H. van

Lenthe, R. Broer, Z. Rashid, submitted] criticized the title

paper[1] on the grounds that (1) the paper neglects to properly

reference the much earlier work by Broer and Nieuwpoort,[2]

who developed an algorithm to use a Fock matrix to compute

a matrix element between two different determinants, and

(2) the paper contains a misleading comparison with van

Lenthe et al.’s valence bond self-consistent field (VBSCF)

algorithm.[3] In addition, these authors made a few more

remarks on our original paper.

First of all, the Broer–Nieuwpoort work is ‘‘concentrated on a

broken symmetry approach to the description of certain exci-

tations and ionizations in systems with spatial symmetry,’’ and

their algorithm for the evaluations of matrix elements between

determinants of nonorthogonal orbitals, presented in the

Appendix, is based on the biorthogonal transformations.[2] In

contrast, our paper discussed an algorithm for the evaluation

of energy gradients, which is used for orbital optimization in

valence bond theory. In the Introduction section of our pa-

per,[1] we have clearly, properly, and correctly referenced the

Broer–Nieuwpoort work (together with a few papers from

other groups) with a comment that ‘‘An alternative method for

the evaluation of Hamiltonian matrix elements of nonorthogo-

nal orbitals was devised by performing biorthogonal transfor-

mation of orbitals, and further expressing the formula in terms

of basis functions, and in such a way, the time-consuming in-

tegral transformation from basis functions to orbitals is

avoided.’’ However, in nowhere in the Broer–Nieuwpoort paper,

any algorithm for orbital optimization was discussed or even

mentioned. Although van Lenthe et al.’s current argument that

the Broer–Nieuwport algorithm ‘‘can be used for an orbital

optimization’’ [J. H. van Lenthe, R. Broer, Z. Rashid, submitted]

seems legitimate, this is apparently out of the scope of our

discussion. Credits are given only when explicit claims with

proofs have been made in the original paper.

Second, the discussion of the scaling of different algorithms

in our paper is valid. For the super-Configuration Interaction

(CI) method, the scaling is O(m2N6), while for the approximate

Newton Raphson (aNR) method, it is O(mN5). For one determi-

nant pair, our algorithm runs at O(m4) for the matrix element.

These numbers are agreed by van Lenthe et al. from their

Comment [J. H. van Lenthe, R. Broer, Z. Rashid, submitted].

However, our algorithm is designed in such a way that the

Fock matrix, constructed from the transition density matrix

between two determinants, can be used to evaluate the first-

order energy derivative with respect to the orbital coefficients

(for more details, see Eq. (31) in our original paper[3]). This is at

the heart of our whole algorithm which consequently scales at

O(m4) for one determinant pair.

It is important to point out that we fully realize the pros

and cons of different algorithms and in our discussion, we

explicitly wrote that ‘‘Since our algorithm needs to build a

Fock matrix for each determinant pair, it is possible that our

algorithm will be less efficient than the aNR method if there

[a] W. Wu

The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian

Provincial Key Laboratory of Theoretical and Computational Chemistry,

College of Chemistry and Chemical Engineering, Xiamen University, Xiamen,

Fujian 361005, China

Fax: (þ86) 592 2184708, E-mail: weiwu@xmu.edu.cn

[b] Y. Mo

Department of Chemistry, Western Michigan University, Kalamazoo,

Michigan 49008

Contract/grant sponsor: National Science Foundation of China;

Contract/grant number: 21120102035; Contract/grant sponsor: Ministry

of Science and Technology of China; Contract/grant number:

2011CB808504; Contract/grant sponsor: US National Science

Foundation; Contract/grant number: CHE-1055310 (to Y.M.); Contract/

grant sponsor: Western Michigan University (FRACAA; to Y.M.).

VC 2012 Wiley Periodicals, Inc.

914 Journal of Computational Chemistry 2012, 33, 914–915 WWW.CHEMISTRYVIEWS.COM

LETTER TO THE EDITOR WWW.C-CHEM.ORG

are too many Slater determinants and m � N.’’[1] It is true that

the real computational costs for aNR/super-CI algorithms may

be reduced by carefully considering the details of a targeted

system, numbers of inactive, active, and virtual orbitals and

imposing the strong orthogonality between inactive and active

orbitals. But what we discussed was the scaling in terms of

the numbers of basis functions and electrons in general cases,

which is an accepted way for methodology developments. In

our algorithm, both the virtual orbitals and Brillouin states are

not required in the evaluations of energy and its gradients. As

for the computational timings listed in our paper (Table 1), we

clearly noted that all valence bond (VB) calculations in the pa-

per were based on our xiamen valence bond (XMVB) pack-

age[4] and no other software was used or even mentioned

with no intention to mislead anyone.

Third, van Lenthe et al. questioned why we did not discuss

the Lagrangian and asserted that ‘‘Without it a proper gradient

optimization is impossible.’’ In our work, however, we actually

used the following expression to derive the energy gradients,

E ¼P

K;L CKCLHKLP

K ;L CKCLMKL¼

PK ;L CKCLMKLðtrPKLhþ trPKLFÞ

2P

K ;L CKCLMKL(1)

In other word, the normalization condition was explicitly

considered in the expression for gradients. Thus, there is really

no need to introduce the Lagrangian constraint in our

algorithm.

In our paper, we directly expressed the energy gradients

with respect to orbital coefficients in terms of basis function-

based quantities, such as basis function integrals and density

matrix. As a remarkable advantage, no integral transformation

from basis functions to orbitals is needed in our algorithm, as

van Lenthe et al. mentioned in their Comment. Even more, no

virtual orbtials and Brillouin states are needed in our method

as well. It is thus unclear for us why van Lenthe et al. counted

the number of Brillouin states needed in our algorithm.

As for the timing comparison between the four-index trans-

formation and the matrix element evaluation, it is hard to say

that the former is always cheaper than the latter except for

the super-CI and aNR methods. The computational costs really

depend on the numbers of determinants, basis functions, and

active orbitals. In many practical applications, we usually con-

sider only a very few (e.g., one or two) covalent bonds for VB

calculations but retain all the rest bond orbitals doubly occu-

pied and nonorthogonal. In these cases, the four-index integral

transformation actually becomes more time consuming than

the evaluation of the matrix elements.

Since the energy gradients in our algorithm are analytical,

we are also wondering what makes van Lenthe et al. believe

that the saving of our algorithm is mainly due to the fact that

fewer gradients are calculated.

Certainly we agree that the generalized Brillouin theorem is

only applicable for orthogonal orbitals. It was thus beyond the

scope of our original paper. Instead, we briefly wrote that the

super-CI method is based on the generalized Brillouin

theorem.

As a last remark, we note that there is no place in our paper

suggesting that self-consistent filed (SCF) equations are used

to find optimal orbitals in our algorithm. Rather, we com-

mented that ‘‘The iterative SCF procedure is in fact the analog

in molecular orbital (MO)-based multi-configuration self-con-

sistent field (MCSCF) theory. However, this algorithm is not ef-

ficient enough, and its convergence is often slow and

unreliable.’’

In conclusion, we believe that the criticisms made by van

Lenthe, Broer, and Rashid are unjustified and insignificant. It is

difficult if not impossible to make strict and universal compari-

sons among different algorithms. In our original paper, we

only briefly addressed the approximate scaling in terms of m

and N based on the literature. It should be reminded again

that Brillouin states, virtual orbitals, and integral transformation

are not needed and thus not involved in our algorithm.

Keywords: Valence bond theory; VBSCF; nonorthogonal orbital;

orbital optimization; energy gradient.

How to cite this article: W. Wu, Y. Mo, J. Comput. Chem. 2012, 33,

914–915. DOI: 10.1002/jcc.22923

[1] L. Song, J. Song, Y. Mo, W. Wu, J. Comput. Chem. 2009, 30, 399.

[2] R. Broer, W. C. Nieuwpoort, Theor. Chim. Acta 1988, 73, 405.

[3] F. Dijkstra, J. H. van Lenthe, J. Chem. Phys. 2000, 113, 2100.

[4] L. Song, Y. Mo, Q. Zhang, W. Wu, J. Comput. Chem. 2005, 26, 514.

Received: 5 December 2011Revised: 9 December 2011Published online on 1 February 2012

WWW.C-CHEM.ORG LETTER TO THE EDITOR

Journal of Computational Chemistry 2012, 33, 914–915 915