A unitary transformation approach to avoiding the diagonalization step in density functional theory calculations is developed into an algorithm that can scale linearly with molecular size. For target accuracy of 10−5 in the rms rotation gradient, the average number of matrix multiples required per self-consistent field iteration is between about 35 (STO-3G) and 50 (6-31G**). This compares favorably to the existing canonical purification method. Crossovers with direct diagonalization are demonstrated for 1D alkane chains and 2D water clusters.

1.
A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (McGraw–Hill, New York, 1993).
2.
R.
McWeeny
,
Rev. Mod. Phys.
32
,
335
(
1960
).
3.
R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).
4.
W. Hehre, L. Radom, J. A. Pople, and P. v. R. Schleyer, Ab Initio Molecular Orbital Theory (Wiley, New York, 1986).
5.
J. M.
Millam
and
G. E.
Scuseria
,
J. Chem. Phys.
106
,
5569
(
1997
).
6.
P. E.
Maslen
,
C.
Ochsenfeld
,
C. A.
White
,
M. S.
Lee
, and
M.
Head-Gordon
,
J. Phys. Chem. A
102
,
2215
(
1998
).
7.
W. Yang and J. M. Pérez-Jordá, in Encyclopedia of Computational Chemistry, edited by P. v. R. Schleyer (Wiley, New York, 1998), pp. 1496–1513.
8.
G. E.
Scuseria
,
J. Phys. Chem. A
103
,
4782
(
1999
).
9.
S.
Goedecker
,
Rev. Mod. Phys.
71
,
1085
(
1999
).
10.
C. A.
White
,
B. G.
Johnson
,
P. M. W.
Gill
, and
M.
Head-Gordon
,
Chem. Phys. Lett.
230
,
8
(
1994
).
11.
C. A.
White
,
B. G.
Johnson
,
P. M. W.
Gill
, and
M.
Head-Gordon
,
Chem. Phys. Lett.
253
,
268
(
1996
).
12.
C. A.
White
and
M.
Head-Gordon
,
J. Chem. Phys.
105
,
5061
(
1996
).
13.
M. C.
Strain
,
G. E.
Scuseria
, and
M. J.
Frisch
,
Science
271
,
51
(
1996
).
14.
M.
Challacombe
and
E.
Schwegler
,
J. Chem. Phys.
106
,
5526
(
1997
).
15.
Y.
Shao
and
M.
Head-Gordon
,
Chem. Phys. Lett.
323
,
425
(
2000
).
16.
E.
Schwegler
and
M.
Challacombe
,
J. Chem. Phys.
105
,
2726
(
1996
).
17.
E.
Schwegler
,
M.
Challacombe
, and
M.
Head-Gordon
,
J. Chem. Phys.
106
,
9708
(
1997
).
18.
E.
Schwegler
and
M.
Challacombe
,
J. Chem. Phys.
111
,
6223
(
1999
).
19.
E.
Schwegler
and
M.
Challacombe
,
Theor. Chem. Acc.
104
,
344
(
2000
).
20.
C.
Ochsenfeld
,
C. A.
White
, and
M.
Head-Gordon
,
J. Chem. Phys.
109
,
1663
(
1998
).
21.
J. C.
Burant
,
G. E.
Scuseria
, and
M. J.
Frisch
,
J. Chem. Phys.
105
,
8969
(
1996
).
22.
J. M.
Pérez-Jordá
and
W.
Yang
,
Chem. Phys. Lett.
241
,
469
(
1995
).
23.
B. G. Johnson, C. A. White, Q. Zhang, B. Chen, R. L. Graham, P. M. W. Gill, and M. Head-Gordon, in Recent Developments in Density Functional Theory, edited by J. M. Seminario (Elsevier Science, Amsterdam, 1996), Vol. 4.
24.
R. E.
Stratmann
,
G. E.
Scuseria
, and
M. J.
Frisch
,
Chem. Phys. Lett.
257
,
213
(
1996
).
25.
J. W. Demmel, Applied Numerical Linear Algebra (Siam, Philadelphia, 1997).
26.
F.
Mauri
,
G.
Galli
, and
R.
Car
,
Phys. Rev. B
47
,
9973
(
1993
).
27.
P.
Ordejón
,
D. A.
Drabold
,
M. P.
Grumbach
, and
R. M.
Martin
,
Phys. Rev. B
48
,
14646
(
1993
).
28.
J. J. P.
Stewart
,
Int. J. Quantum Chem.
58
,
133
(
1996
).
29.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).
30.
C.
Ochsenfeld
and
M.
Head-Gordon
,
Chem. Phys. Lett.
270
,
399
(
1997
).
31.
T.
van Voorhis
and
M.
Head-Gordon
,
Mol. Phys.
100
,
1713
(
2002
).
32.
B.
Dunietz
,
T.
van Voorhis
, and
M.
Head-Gordon
,
J. Theor. Comput. Chem.
1
,
255
(
2002
).
33.
S.
Goedecker
and
L.
Colombo
,
Phys. Rev. Lett.
73
,
122
(
1994
).
34.
R.
Baer
and
M.
Head-Gordon
,
J. Chem. Phys.
107
,
10003
(
1997
).
35.
A. H. R.
Palser
and
D. E.
Manolopoulos
,
Phys. Rev. B
58
,
12704
(
1998
).
36.
X. P.
Li
,
R. W.
Nunes
, and
D.
Vanderbilt
,
Phys. Rev. B
47
,
10891
(
1993
).
37.
M.
Challacombe
,
J. Chem. Phys.
110
,
2332
(
1999
).
38.
As pointed out by Helgaker, Larsen, Olsen, and Jørgensen in Ref. 40, one can also parameterize the Hartree–Fock/Kohn–Sham energy this way, leading to a diagonalization-free approach. Here the most desirable minimization method is BFGS, because it does not require extra Fock builds to find the next step direction. But parallel transport of previous gradient and step vectors is not quite straightforward even though its MO counterpart was already developed in Refs. 31 and 32.
39.
T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, New York, 2000).
40.
T.
Helgaker
,
H.
Larsen
,
J.
Olsen
, and
P.
Jo/rgensen
,
Chem. Phys. Lett.
327
,
397
(
2000
).
41.
H.
Larsen
,
J.
Olsen
,
P.
Jørgensen
, and
T.
Helgaker
,
J. Chem. Phys.
115
,
9685
(
2001
).
42.
M.
Head-Gordon
,
Y.
Shao
,
C.
Saravanan
, and
C. A.
White
,
Mol. Phys.
101
,
37
(
2003
).
43.
G. H.
Weiss
and
A. A.
Marududin
,
J. Math. Phys.
3
,
771
(
1962
).
44.
C. A.
White
,
P. E.
Maslen
,
M. S.
Lee
, and
M.
Head-Gordon
,
Chem. Phys. Lett.
276
,
133
(
1997
).
45.
As pointed out by Challacombe in Ref. 37, the inverse overlap matrix is itself dense, but one can avoid building it explicitly by using the inverse Cholesky factor, Z, which is not dense.
46.
C. Saravanan, Y. Shao, R. Baer, P. N. Ross, and M. Head-Gordon, J. Comput. Chem. (in press).
47.
Y. Saad, Version 2.0, Department of Computer Science and Engineering, University of Minnesota, http://www.cs.umn.edu/Research/arpa/SPARSKIT/sparskit.html (unpublished).
48.
J.
Kong
,
C. A.
White
,
A. I.
Krylov
et al.,
J. Comput. Chem.
21
,
1532
(
2000
).
49.
P.
Pulay
,
Chem. Phys. Lett.
73
,
393
(
1980
).
50.
P.
Pulay
,
J. Comput. Chem.
3
,
556
(
1982
).
51.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
).
52.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
53.
We believe the explanation for why Ref. 41 required projection of redundant variables and our approach does not is not because of our use of an orthogonal basis, vs the nonorthogonal basis of Ref. 41. In an orthogonal basis, there is no distinction between covariant and contravariant vectors, and thus a covariant gradient can automatically be converted into a contravariant step which has no component in the occupied–occupiedor virtual–virtual spaces. This applies either to steepest descent or to a full Newton step. It is no longer exactly true when cutoffs are applied or when a level shift is added to the Hessian. However, the latter is only done far from convergence. By contrast, in a nonorthogonal basis, when the inverse metric is not used to convert covariant vectors to contravariant, even a steepest descent step will have artificial components in the occupied–occupied and virtual–virtual spaces, because such a step is not tensorially correct.
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