#MoRiBS-PIMC: Molecular Rotators in Bosonic Solvents with Path-Integral Monte Carlo#

Disclaimer: We disclaim any and all warranties concerning the enclosed program.

The program MoRiBS-PIMC is mainly written by Tao (Toby) Zeng, Nicholas Blinov, Gregoire Guillon, Hui Li, and Pierre-Nicholas Roy at University of Alberta and University of Waterloo, Canada. If you encounter any problem with respect to compiling or running the program, please contact T. Zeng by email ([email protected] or [email protected]).

##Compilation##

Users should not change the directory structure after they unzip the distributed file. We label the main directory, where the source codes (*.cc, *.h, and *.f) are, as $MAIN. Please note that we have provided example configuration files like makefile with the distribution and one may just adjust the files according to the architecture of his/her computer. There is no need to create new configuration files. The compilation procedure of MoRiBS-PIMC contains the following steps:

---

###NOTE### If you clone this repository using Github, you will need to have Git Large File Storage installed in order to download some of the larger files associated with the examples/ directory. Github only permits versioning for files less than 100MB in size. If you have git-lfs installed, when you clone the repository, all of the example files will download properly.

For Ubuntu, you can download the appropriate binary from https://github.com/github/git-lfs/releases and install using the install.sh file. Please refer to https://git-lfs.github.com/ for full details for various operating systems.

---

1. Download and unpack the source code from the repository. This unpacked directory contains the source code and for future reference in this document, we will refer to this directory as $MAIN, where it is understood that $MAIN refers to the path of the directory (i.e. /home/user/MoRiBS-PIMC-master or similar) • Note: The directory structure after unpacking is important and should not be modified by the user. There are appropriate configuration files included in the source code that may be changed slightly to accommodate different architectures.
2. The first configuration is required for Scalable Parallel Random Number Generator, SPRNG. Open $MAIN/sprng/make.CHOICES and specify the platform of the computer by uncommenting the correct line (i.e. PLAT=LINUX)
3. Open $MAIN/sprng/SRC/make.$(PLAT) where $(PLAT) is the same platform specified in Step 2. In make.$(PLAT), specify the location of the Fortran and C/C++ compilers for the com- puter by editing the lines beginning with F77= and CC= respectively. For example, for the standard compilers within Ubuntu, one should specify F77 = /usr/bin/gfortran and CC = /usr/bin/gcc • Note: Only the non-MPI compilation of SPRNG has been tested with MoRiBS-PIMC .
4. Within the $MAIN/sprng/SRC directory, execute make clean and then make. If make finishes without error, SPRNG has been successfully compiled. • Note: Ensure $MAIN/sprng/SRC/lib is empty before running make.
5. To compile the rest of MoRiBS-PIMC , navigate to the $MAIN/ directory and enter ./configure which should locate your C++ and Fortran compilers. • Note: If you would like to use specific compilers, you can specify them via ./configure CXX=/usr/bin/g++ FC=/usr/bin/gfortran or similar.
6. Within the $MAIN directory, execute make clean and then make. If there are no error mes- sages, an executable file is generated called pimc in $MAIN and you have successfully compiled MoRiBS-PIMC .
7. In order to verify the installation, the user may run some of the examples included with the source code. From the $MAIN directory, execute the following commands: • cd examples/MF 8He 0.37K 512 128/ • cp ../../pimc . • ./pimc If MoRiBS-PIMC was compiled correctly, the example should run without issue. The details regarding the input files are provided in the publication. • Note: The files with the name yw001.* or permutation need to be deleted prior to running ./pimc. These files are generated by pimc but need to be deleted and regenerated everytime pimc is executed.

##BEFORE RUNNING##

For full information of running this program, please read our associated paper, whose reference will be posted once it is published. Given any problems at this time being, please contact [email protected].

Please note that one should delete the following files before each simulation run: 1: yw001.*; 2: permutation, Otherwise, the simulation will bomb out.

Remember to use asymrho.x (in nmv_prop/), symrho.x (in symtop_prop/) or linden.x (in linear_prop/) to generate the needed files for PIMC sampling of asymmetric top, symmetric top, or linear rotors. Users should read the respective README files in those directories carefully before compiling and running those programs. Users should carefully name the resultant files in accordance to the rules explained in our CPC paper.

Users also need to prepare potential files for particle-particle and rotor-particle interactions. A particle-particle potential file should have the following format:

#  Aziz potential (aziz-2) (PRL 74 ,1568 (1995))
#
# (\AA)   V(r) (K)
   1    45932.4
  1.05812    36300.3
   1.11623    28636.2
   1.17435    22545.8
   ...           ...
    29.7094    -1.47775e-05
    29.7675    -1.46051e-05
    29.8257    -1.4435e-05
    29.8838    -1.42672e-05
    29.9419    -1.41017e-05
    30    -1.39384e-05

There can be as many comment lines starting with # before the first row of actual data, but not between the data lines.

A linear rotor-particle potential file should have the following format:

        2001        1001   # number of radial and polar angle grid points
 0.0050  0.0020   #  intervals of r and cos(theta)
   2.0000  # first r value
   2.0050
   2.0100
   2.0150
   ....
  11.9800
  11.9850
  11.9900
  11.9950
  12.0000  # last r value
  -1.0000  # first cos(theta) value
  -0.9980
  -0.9960
  -0.9940
  -0.9920
   ....
   0.9920
   0.9940
   0.9960
   0.9980
   1.0000  # last cos(theta) value
   86835.25007847  # first potential in K
   85831.53310409
   84866.79241540
   83938.89463026
   83045.83110925
   ....
      -0.05370867
      -0.05380450
      -0.05390041
      -0.05399636
      -0.05409233
      -0.05418828  # last potential in K

Note that all those # comments should be removed in an actual potential file. The potentials should be arranged in the following order:

	loop over r grid
		loop over cos(theta) grid

A non-linear rotor-particle potential file should have the following format:

501 181 181 4.0 20.0  # number of radial, polar angle, and azimuthal angle grid points, initial and final r values. Note that the number of the polar angle grid points should be fixed to be 181, meaning from 0 to 180 degrees. The number of the azimuthal angle grid points can be 91, 181, and 361, depending on the symmetry of the rotor. For a C2v type rotor, 91 is fine. For a Cs type rotor, 181 is fine. For most rotors, 361 should be used. Note that 181 is coded only for Cs type rotor. A C2 type rotor still requires 361.
 0.95643446E+03  # the first potential in K
 0.95643446E+03
 0.95643446E+03
 0.95643446E+03
 0.95643446E+03
 ....
-0.14296732E+00
-0.14296732E+00
-0.14296732E+00
-0.14296732E+00
-0.14296732E+00
-0.14296732E+00
-0.14296732E+00  # the last potential in K

The potentials should be arranged following

	loop over radial grid
		loop over polar angle grid
			loop over azimuthal angle grid

Six examples of input are provided in the examples/ directory. They are: 1. H2Odimer_0.74K_4096_2048: H2O dimer at T = 0.74 K with 4096 translational and 2048 rotational beads; 2. MF_1He_0.37K_512_128: Methyl Format and He at T = 0.37 K with 512 translational and 128 rotational beads; 3. MF_8He_0.37K_512_128: Similar to the previous example but with 8 He atoms; 4. N2O_5pH2_0.5K_512_128: N2O with 5 para-H2 at T = 0.5 K with 512 translational and 128 rotational beads; 5. SO2_1pH2_0.37K_1024_256: SO2 with 1 para-H2 at T = 0.37 K with 1024 translational and 256 rotational beads; 6. SO2_4pH2_0.37K_1024_256: Similar to the previous example but with 4 para-H2 molecules.

One may compile the code, move the resultant executable pimc to examples/MF_8He_0.37K_512_128 or examples/N2O_5pH2_512_128, and try running the respective simulations. One may also look at the files in those two example directories to have a sense of how many files are needed and their formats. Some of these examples are discussed in our associated paper.

##OUTPUT SUMMARY##

There are two relevant directories for a MoRiBS-PIMC run, the working directory and the output directory. The former is the directory where the program executable is and the latter is specified in qmc.input. In below we simply call the two directories $WORK and $OUTPUT. And we also use $FILNAM to specify the FILENAMEPREFIX that is specified in qmc.input.

###Energy### All energy outputs are stored in the following two files:

```$OUTPUT/$FILNAM_sum.eng and $OUTPUT/$FILNAM.eng```.

The first contains the energy components averaged on the fly while the latter contains the block-averaged quantities for every block. In $OUTPUT/$FILNAM_sum.eng, column 1: counting indices of punching out the energy components; 2: translational kinetic energy; 3: potential energy; 4: the sum of translational and potential energies; 5: rotational kinetic energy; 6: heat capacity term A (CvA) for debugging; 7: total energy (translational + potential + rotational); 8: total heat capacity; 9: heat capacity term B (CvB) for debugging; 10: heat capacity term C (CvC) for debugging.

All energy quantities have the unit of Kelvin (K) and the heat capacity has the unit of Kb, which is the Boltzmann constant. The CvX quantiites are not of interest for general users. Interested users can look at the SaveSumEnergy routine in mc_main.cc to figure out what they are. CvB should be equal to the pure translational heat capacity when the translational motion is not coupled to any potential. CvC should be equal to the pure rotational heat capacity when the rotational motion is not coupled to any potential.

In $OUTPUT/$FILNAM.eng, column 1: block index; 2: block averaged translational energy; 3: block averaged potential energy; 4: block averaged translational + potential energies; 5: block averaged rotational energy; 6: heat capacity term for debugging; 7: block averaged total energy (translational + potential + rotational); 8: heat capacity related term (CvE) for calculating averaged Cv; 9 and 10: heat capacity terms for debugging.

All energy quantities have the unit of K while CvE has the unit of K^2. The three debugging terms are not of general interest and not explained here. Users can calculate the averaged quantities and variances by treating each block averaged value as an observed result. Thus, the number of block has the meaning of the number of observation. The Averaged Cv should be calculated as

            [ 1.5*N*P*T - < E > ]^2 + < CvE >
< Cv > = -----------------------------------
                             T^2

N is the total number of particles, including point-like particles and rotors, P the number of translational beads, and T the temperature in Kelvin. The resultant < Cv > has a unit of Kb, and its variance is obtained through error propagation.

###Distribution###

Whenever there are point-like particles, the $OUTPUT/$FILNAM_sum.gra contains pair radial distribution between point-like particles. The first column has radius and the second the rho(r). It follows the convention that integrating r from 0 to infinity rho(r)*dr gives 1. In practice, the density is made by binning in a finite grid, which is also normalized to 1.

If there is a linear rotor, the $OUTPUT/$FILNAM_sum.g2d file contains the 2-D distribution of the point-like particles around the rotor. The 2 dimensions are radius and polar-angle. The file has three columns, where column

1: polar angles in degree;
2: radius in Angstrom;
3: density rho(r, theta).

The density follows the following normalization:

Integrating from 0 to pi for theta and from 0 to infinity for r rho(r,theta)dr dtheta = 1.

For a non-linear rotor, there are three files:

	$OUTPUT/$FILNAM_sum.gri;
	$OUTPUT/$FILNAM_sum.grt;
	$OUTPUT/$FILNAM_sum.grc

which contain the radial (r), polar angle (theta), and azimuthal angle (chi) distributions of the point-like particles in the molecu-fixed frame (MFF) of the rotor. The MFF is the same as the one used in the definition of Eular angles. The densities follow the following normalizations:

Integrating r from 0 to infinity rho(r) dr = number of point like particles;
Integrating theta from 0 to pi rho(theta) dtheta = 1;
Integrating chi from 0 to 2*pi rho(chi) dchi =1.

In addition, at the beginning of each block, the program punches out a $OUTPUT/$FILNAM###.xyz file to record the path configuration of the system. The file has N*P+2 lines, where N is the number of all particles (point-like + rotor) and P is the number of translational beads. Line one contains the permutational table for the configuration snapshot and line two is a comment line. The rest lines loop over all particles, point-like first and rotor second, and for each particle loop over all beads and punch out the corresponding configurations. Column 1 has the particle name, columns 2, 4, and 6 the x, y, and z coordinates in the space-fixed frame respectively, and columns 3, 5, 7 the phi, cos(theta), and chi angle coordinates. For linear rotors, phi and theta are the azimuthal and polar angles. For a top, phi, theta, and chi are the Eular angles defined in Figure 3.1, page 78 of Zare (Angular Momentum, 1988, John Wiley & Sons, Inc). When the number of rotational beads (Q) is a fraction of P, only the first Q lines of each particle block contains the meaningful angular coordinates.

###Superfluidity###

When bosons are involved, the program created a $OUTPUT/$FILNAM.sffs3d file to record the block-averaged classical moment of inertia of the bosons in the space-fixed frame (SFF) and the quantum deductions from area estimator. Readers should consult PRL, 1989, 63, 1601-1604 for more details of the area estimator. This estimator gives

I_{ij}^{eff} = I_{ij}^{cl} - 4m^2 < A_i A_j > / ( beta*hbar^2 ),

where the subtrahend is the quantum deduction. In $OUTPUT/$FILNAM.sffs3d, column

1: block indices;
2 - 10 : the xx, xy, xz, yx, yy, yz, zx, zy, and zz elements of the block-averaged classical moment of inertia;
11 - 16 : the xx, xy, xz, yy, yz, and zz components of the quantum deduction tensor.

All quantities above have the unit of u*Angs^2. If the simulation is ergodic, the off-diagonal elements of the I^{cl} and the quantum deduction tensor should be averaged to zeros and making the SFF effective moment of inertia to be diagonal.

If a top rotor is involved, then a $OUTPUT/$FILNAM.mffs3d file will be created. This file has the same format as the $OUTPUT/$FILNAM.sffs3d file but with the properties calculated along the molecule-fixed frame (MFF) axes. One can calculate the MFF effective moment of inertia using data in this file. Note that off-diagonal matrix element may occur since the MFF is usually anisotropic.

If a linear rotor is involved, a $OUTPUT/$FILNAM.sup file will be created. It has the following content: Column 1: block indices; Column 2: block-averaged superfluid fraction perpendicular to the rotor axis; Column 3: block-averaged superfluid fraction parallel to the rotor axis; Column 4: block-averaged classical moment of inertia perpendiculal to the rotor axis; Column 5: block-averaged classical moment of inertia parallel to the rotor axis; Column 6 and 7: debugging data.

Note that averaged fraction of column 2 multiplied by the averaged moment of inertia of column 4 gives the effective moment of inertia I_b of the effective symmetric top rotor and can be used to calculate the rotational constant B. Similarly, the non-degenerate rotational constant (A or C depending on prolate- or oblate-like) of the effective symmetric top rotor can be obtained from averaged quantities of columns 3 and 5.

When one rotor is involved in the simulation, no matter it is a linear or a top, the program creates a $OUTPUT/$FILNAM_sum.rcf file, which records the averaged imaginary time orientational correlation function for the z-axis of the MFF. This function can be used to extract effective rotational constant of the effective rotor. Please refer to JCP, 2004, 120, 5916-5931 for details.

Another superfluidity-relevant file is the $WORK/permutation.tab file. It records the permutation table of all the bosons from time to time and it can be used to calculate the probability of a boson being in a permutation cycle with a certain length.