
The pair density matrices are calculated using the matrix squaring method described in section 2.3.1. They are stored in tables using the expansion formula Eq. 2.38 and then entered into the PIMC simulation program. The accuracy of these tables is crucial for all following results. Using the precomputed pair density matrices allows one to employ a much larger time step because one starts with a solution of the twoparticle problem. Fig. shows how accurate this method is. The internal energy of an isolated hydrogen atom at sufficiently temperature ( ) in a large box () is compared with the exact groundstate energy of . The temperature was chosen low enough so that excited states can be neglected e.g. the contribution to the energy from the occupation of first excited state is at this temperature. Furthermore, it is tested whether the kinetic energy and the potential energy satisfy the virial theorem . If only diagonal action terms are considered in Eq. 2.38 one finds a rather slow convergence as function of the number of time slices (Fig. ). Eventually, the error goes to zero in the Trotter limit, Eq. 2.24, of an infinite number of slices. Using offdiagonal terms in the expansion formula, Eq. 2.38, improves the convergence significantly as shown in Fig. . One can use different orders to calculate the action and the energy . The resulting accuracy from different orders is shown in Tabs. and . It reveals that using order 2 or higher instead of order 1 in the action decreases the errors by almost one order of magnitude. This is an important observation because most manyparticle simulations reported in this work had been performed with and (which was found to be sufficient for simulations of Helium particles) before this analysis was done. Because of the resulting inaccuracies, the estimated energies and pressures are slightly too high.
energy  action order  action order  
order  0  1  2  3  0  1  2  3 
0  0.8(1)  1.74(3)  
1  0.47(2)  0.18(3)  0.071(25)  0.176(08)  0.074(11)  0.043(07)  
2  0.56(2)  0.07(3)  0.013(48)  0.166(05)  0.063(09)  0.034(13)  
3  0.61(4)  0.06(3)  0.039(08)  0.183(11)  0.056(11)  0.031(03) 
0.5  1  2  0.5  1  2  
7812  0.41(3)  0.16(3)  0.06(8)  0.422(10)  0.112(8)  0.030(22) 
3906  0.41(3)  0.19(3)  0.14(4)  0.419(07)  0.110(5)  0.008(11) 
1953  0.43(3)  0.16(2)  0.08(3)  0.421(12)  0.115(9)  0.009(09) 

(198) 
The accuracy of PIMC simulations of an isolated molecule is affected by the order in action and energy expansion as well as by the time step because it is a fourparticle problem. Tab. shows results calculated with orders. First, we studied the different temperatures and found no dependence on , which means that contributions from electronic excited states are negligible. Furthermore, the comparison of different time steps shows a significant dependence. Using a time step allows one to calculate the energy with an accuracy of approximately eV and with an error of about eV per atom. Using a smaller time step would bring the results in Fig. closer to the exact results.
Most manyparticle PIMC simulations discussed in the following
sections have been performed using and in the
action and energy expansion of the pair density matrices. The correction
resulting from higher order terms will be estimated based on the
following argument. Higher order offdiagonal terms are large for
small separations of the two particles. Therefore, we expected the
dominant corrections to come from pairs of protons and electron when
both particles are close together. Therefore, we suggest to use the integral of
the protonelectron pair correlation function up to a cutoff radius,
(199) 