An efficient implementation of Slater-Condon rules

Anthony Scemama, Emmanuel Giner

Slater-Condon rules are at the heart of any quantum chemistry method as they allow to simplify 3N-dimensional integrals as sums of 3- or 6-dimensional integrals. In this paper, we propose an efficient implementation of those rules in order to identify very rapidly which integrals are involved in a matrix element expressed in the determinant basis set. This implementation takes advantage of the bit manipulation instructions on x86 architectures that were introduced in 2008 with the SSE4.2 instruction set. Finding which spin-orbitals are involved in the calculation of a matrix element doesn't depend on the number of electrons of the system.

http://arxiv.org/abs/1311.6244

Benchmarks

The Intel Fortran compiler and the GNU C compiler are needed.

To run the benchmarks, run

make tests

This command will build the four benchmarks:

• test_cu_avx
• test_cu_sse2
• test_h2o_avx
• test_h2o_sse2

The output should look like this:

==============
Cu, SSE2
==============
taskset -c 0 ./test_cu_sse2
 ndet =        10000
 Cycles n_excitations:   34.7540077400000       +/-   3.479021910893892E-005
 CPU    n_excitations:   1.12000000000000     
           0
 Cycles get_excitation:   40.2044544800000       +/-   4.030457234270350E-005
 CPU    get_excitation:   1.29000000000000     
 Cycles zero   :   30.4083702667599       +/-   1.008247307982627E-004
 Cycles single :   87.7390566250000       +/-   1.829049539625737E-003
 Cycles double :   144.569855767367       +/-   1.169988940582419E-004
 Cycles other  :   30.2330823827413       +/-   1.726071165104671E-005
 Cycles density matrix :    38.7053945194519     
 CPU    density matrix :   0.620000000000001     
==============
Cu, AVX
==============
taskset -c 0 ./test_cu_avx
 ndet =        10000
 Cycles n_excitations:   5.10230640000000       +/-   5.165095393066429E-006
 CPU    n_excitations:  0.170000000000000     
           0
 Cycles get_excitation:   8.66424608000000       +/-   8.905112566651616E-006
 CPU    get_excitation:  0.280000000000000     
 Cycles zero   :   4.93462403836547       +/-   1.572017678605744E-005
 Cycles single :   46.2361855000000       +/-   9.642070951752019E-004
 Cycles double :   78.7195363408724       +/-   6.373098925677101E-005
 Cycles other  :   5.54228558173953       +/-   3.202738133568022E-006
 Cycles density matrix :    6.53137199719972     
 CPU    density matrix :   0.110000000000000     
==============
H2O, SSE2
==============
taskset -c 0 ./test_h2o_sse2
 ndet =        10000
 Cycles n_excitations:   72.4396000700000       +/-   7.250726495869192E-005
 CPU    n_excitations:   2.34000000000000     
           0
 Cycles get_excitation:   81.2813032900000       +/-   8.146127076930712E-005
 CPU    get_excitation:   2.62000000000000     
 Cycles zero   :   58.5725555999600       +/-   1.862427473250057E-004
 Cycles single :   126.333416652778       +/-   1.758400588223553E-003
 Cycles double :   194.749878845180       +/-   1.650472976311530E-004
 Cycles other  :   63.7100840413763       +/-   3.530620172058838E-005
 Cycles density matrix :    76.8598384438444     
 CPU    density matrix :    1.24000000000000     
==============
H2O, AVX
==============
taskset -c 0 ./test_h2o_avx
 ndet =        10000
 Cycles n_excitations:   10.1925470600000       +/-   1.034849403182130E-005
 CPU    n_excitations:  0.330000000000000     
           0
 Cycles get_excitation:   18.1355976500000       +/-   1.843817205266690E-005
 CPU    get_excitation:  0.590000000000000     
 Cycles zero   :   6.19700369667299       +/-   1.990164909991091E-005
 Cycles single :   51.8508057777778       +/-   7.221451379202684E-004
 Cycles double :   88.6343955596826       +/-   7.506068582786541E-005
 Cycles other  :   6.70027751963713       +/-   3.742039226306923E-006
 Cycles density matrix :    11.6624591059106     
 CPU    density matrix :   0.190000000000000   

Source files:

• nsubst1.x.f90 : Find the number of substitutions between two determinants
• exc0.x.f90 : Dispatch function for getting the excitation operator from determinants
• exc1.x.f90 : Finds the single excitation operator
• exc2.x.f90 : Finds the double excitation operator
• density_matrix.x.f90 : Computation of the one-electron density matrix
• rdtsc.c : Get number of CPU cycles from hardware counter

Data Files

• cu.dat : 10 000 determinants obtained from CIPSI in cc-PVDZ basis set for the Cu atom
• cu.coef : CI Coefficients
• h2o_determinants.dat : 10 000 determinants obtained from CIPSI in cc-PVTZ basis set for the water molecule