Iterative Complement (IC) method for solving the Schrödinger
equation of Atoms and Molecules
T. Fanga) and H. Nakatsujia)a) Quantum Chemistry Research Institute, Kyodai Katsura Venture Plaza North,Goryo Oohara 1-36, Nishikyo-ku, Kyoto 615-8245, JapanThe Iterative Complement (IC) method,1,2 was presented to solve the Schrödinger
equations of atoms and molecules. In this method, the Hamiltonian of a system is divided into
ND parts, and the wave function is defined in the recurrence relationé NDùy n +1 = ê1 + å C I , n g ( H I - E I , n ) ú y n
ë I =1û

where E I , n is defined by y n g ( H I - EI , n ) y n = 0 . The corresponding set of secularequations is solved to determine the ND variables. Compared with the Free Complement (FC)method,3 the IC method has the merit that the total number of variables is much smaller.
Moreover, the dimension of the matrix does not increase as the computation order grows. So
it is essentially a diagonalization-free method and is very suitable to super-parallel computers,
though the IC method converges more slowly than the FC method.
In order to increase the convergence speed, we have developed the method which
re-optimizes the lower order coefficients with the existence of the higher order variables. Theinteraction among complement functions from different orders is explicitly taken into account.
We have formulated this procedure in the iterative manner. Accordingly the secular equations
are solved iteratively and the convergence of the IC method is considerably improved.
To further increase the convergence, we introduce more freedom to the IC wave function.
We combine the calculation of different levels together, and then all the independent operators
are collected to generate complement functions. The number of variables in each iteration islarger compared to the original IC method because the Hamiltonian is divided to themaximum partition, so that the IC wave function is more relaxed. While the dimension ofmatrix in each iteration step is still constant, so the computational effort for thediagonalization does not increase as the order grows.
Table I: Energies of Original IC method, relaxed IC method, extended IC method, andFC method. Helium Atomextended-IC2
FCOriginal IC relaxed ICOrder
Total
number(ND=1) (ND=1) MN
EnergyMNEnergyof variables1 í2.8983832 í2.8998717
6
í2.9015710
2 í2.9009105 í2.9024739 26
26
í2.9036966 26
í2.9036966
3 í2.9015583 í2.9032702
74
í2.9037224

4 í2.9018020 í2.9035185 57
83
í2.9037232 159 í2.9037240
5 í2.9019321 í2.9036142
291 í2.9037243

6 í2.9020136 í2.9036580 57
140
í2.9037241 481 í2.9037243
7 í2.9020695 í2.9036809
738 í2.9037243
8 í2.9021102 í2.9036944 57
197
í2.9037242 1074 í2.9037243
Exact Valueí2.903724377

The IC method with variational principle is demonstrated to be very effective, however,
the analytical integrals are not always feasible. So the Local Schrödinger equation (LSE)
method,4 is combined with the IC method to solve the Schrödinger equation of general atoms
and molecules. Moreover, The combination of LSE with the IC method is especially suitablefor parallel computation, because the IC method avoided large dimensional matrix-matrix
multiplication, and the sampling itself is a series of independent calculation which can beparalleled effectively.
In contrast to the FC method, the IC method is essentially a single state theory, so theexcited states are not obtained simultaneously with the ground state. So the IC-configurationinteraction (CI) method is presented to calculate excited states from the IC ground state wave
function. The frame work of the IC-CI method is similar to the SAC-CI method proposed in
1978. The IC-CI SD and IC-CI general approaches are formulated.2,5Reference:[1] H. Nakatsuji, J. Chem. Phys., 113, 2949 (2000).
[2] H. Nakatsuji and E. R. Davidson, J. Chem. Phys., 115, 2000 (2001).
[3] H. Nakatsuji, Phys. Rev. Lett., 93, 030403 (2004).
[4] H. Nakatsuji, H. Nakashima, Y. Kurokawa, and A. Ishikawa, Phys. Rev. Lett., 99, 240402(2007).
[5] H. Nakatsuji, J. Chem. Phys., 116, 1811 (2002).