Benzene Dimer: dispersion forces and electronic correlation

Transcription

1 Benzene Dimer: dispersion forces and electronic correlation Introduction The Benzene dimer is an ideal example of a system bound by π-π interaction, which is in several cases present in many biologically relevant systems (stacking interactions in DNA base pairs, aromatic side-chain interactions in proteins). An accurate description of the π-π interaction is in general a challenging task for the computational methods, since noncovalent interaction between benzene rings results in a very shallow potential energy surface (PES). Such interaction in benzene dimers has mainly three components: electrostatic quadrupolequadrupole interactions (dipolar terms are obviously vanishing for symmetry reasons) - attractive or repulsive; dispersion forces - attractive, such as London s; exchange repulsion energy [1]. The subtle interplay between such components results into few minimal-energy dimer arrangements, the most important ones are known as T-shaped and parallel displaced (PD). Both T-shaped and PD dimers are found in both gas-phase and solid-state structures, with a larger stability of the T-shaped in the first case and a larger abundance of the PD in the latter. Being more relevant for the analogies with biological systems, we will mainly concentrate today on the parallel displacement (PD) dimer [2]. Due to such complex effects, this is an excellent test case for comparing different computational methods. We will test three levels of theory: 1. B3LYP is one of the most common DFT functionals, widely used in the physical-chemistry literature for many types of calculations. It is based on an approximate treatment of the exchangecorrelation energy, so that in some cases such a functional could be inappropriate. 2. a simple, empirical (and computationally cheap) correction to B3LYP for the dispersion interactions has been introduced by Grimme, and the corrected functional is usually referred to as the B3LYP-D3. 3. a more refined (and more computationally expensive) method, based on the second order Rayleigh-Schroedinger perturbation treatment of the electronic correlation is the Moller-Plesset (MP2) method on the Hartree Fock results. It is worth noting that there exist many other methods able to provide a more accurate treatment of the correlation, rooted into wavefunction theories (based on multideterminant expansions of the wavefunction and usually referred to as the post-hartree-fock methods), Green function approaches (GW) and many body theories. Unfortunately, such methods are very demanding from a computational point of view, and for our system we chose the MP2 method as the optimal compromise between the computational feasibility and the accuracy in the treatment of the correlation. Thus, MP2 will be used as reference for the other methods. In this tutorial you will sample the PES of a benzene dimer along the stacking coordinate of the PD dimer, using these three levels of theory (B3LYP, B3LYP-D3 and MP2). Optionally, you could also test the B2PLYP functional, which is an intermediate level of theory between the B3LYP and the MP2 methods. The other coordinate (displacement) could be also investigated optionally. The gaussian basis set which will be used throughout the tutorial is the Pople s 6-31G(d,p) basis set. Such a basis belongs to the family of the double-zeta basis sets, augmented with diffuse (d) and polarization (p) functions. Using such methods, we will calculate the binding energy as a function of the distance between the monomers (the stacking coordinate), and determine which one appears to be the most appropriate computational method for this system and why. The binding energy (ΔE bind ) as a function of the intermolecular distance (R) can be evaluated with the supramolecular approach, as the difference between the total energy of the dimer and the total energy of the two infinitely distant monomers ΔE bind (R) = E dimer (R) - 2 E momer Generally speaking, a sampling of a molecular PES could be performed either by a series of singlepoint calculations on user-given input geometries, or in an automatized way, implemented in gaussian by the scan keyword. For the latter, the coordinate R corresponding to PES sampling must be defined in internal coordinates in the input file). In this tutorial we will follow the first approach for

2 didactic purposes, and an expert user could still automatize the input preparation by the knowledge of the shell commanding language (i.e. bash). As a final remark, it is worth noting that the binding energies at any point of the PES should be corrected for the Basis Set Superposition Error (BSSE) [1], which is a source of error occurring whenever fragmented systems are treated, due to the finite basis set size. Such an error is usually small when large basis sets are used (possibly including polarization and diffuse functions). In this tutorial we will neglect BSSE: for further reading see [1,3,4]. Procedure Setup: login to vm-linux-cecam-ad.sissa.it with your yoursissausername ssh X yoursissausername@vm-linux-cecam-ad.sissa.it A. Perform a scan along the intramolecular distance of the benzene dimer - part 1 1. in the /scratch/qm_school folder you will find a directory benzene_dimer with the files needed for this exercise Inside the directory you will find a coordinates file (benzene_eq.xyz) in which you have the equilibrium position (optimized geometry) of the parallel displaced benzene dimer obtained at the MP2 level of theory. 2. In this geometry the two benzene rings lie at different z values. All the atoms in the first benzene monomer have the same quota (z=0.0). You can generate different geometries by manually varying the z coordinate of all the atoms of that monomer. We suggest to generate 5 geometries by increasing z (i.e. reducing distance between monomers) with a step of 0.2 Angstrom, and 7 geometries by decreasing z (i.e. increasing distance between monomer) by 0.2 Angstrom and save each geometry in the xyz format. Eventually, you will have 13 xyz files (one file for the equilibrium position z=0, then 5 files with positive z values, 7 files with negative z values). You could later on extend this scan (see OPTIONALS). 3. Copy (or rename) each xyz file giving the gaussian input extension (com) cp geometry_1.xyz geometry_mp2_1.com 4. Open each com file and add the specifications for the number of processors to be used, memory, and methods. Each gaussian input file should appear as %nprocs=4! number of processors %mem=2gb! memory %chk=geometry_mp2_1.chk! a binary repository where the data are saved #p MP2/6-31G(d,p)! method/basis set scf=tight int=ultrafine! convergence criteria! blank line benzene dimer with z=0.4! comment line! blank line 0 1! total charge and spin multiplicity C ! atomic label and cartesian coordinates of all the atoms [...]! [other coordinate lines]! blank line (DON T FORGET this one!!) Just use the B3LYP (and B2PLYP) keywords in place of MP2 when you create the input files for B3LYP and B2PLYP methods, respectively. For the B3LYP-d3 method, the fourth line should be #p B3LYP/6-31G(d,p) EmpiricalDispersion=GD3 We suggest to create a folder relative to each method you are using with the mkdir command, e.g.

3 mkdir b3lyp In case you want a better explanation of the keyword you are using have a look at: For each level of theory you need to perform also a single point calculation on the benzene monomer (see the formula in the introduction). To do so, create one gaussian input file for the monomer for each method. The geometry of the monomer can be taken from the starting file we provided (benzene_eq.xyz), by considering only the first 12 atoms (first monomer). 6. Copy the input files you have generated on the cluster in which you will run the calculations, e.g. scp geometry_mp2_1.com yourcinecausername@login.galileo.cineca.it: in the same folder we have provided you there is also a job submission script g09.sh. Copy also this file to the cluster: scp g09.sh yourcinecausername@login.galileo.cineca.it: Now you will find in the home directory of the clusters the geometries on which to run the calculations and g09.sh file which is the script file to send the calculations on the cluster. In the following we will use XX=B3LYP, MP2, etc. to mean the string equivalent to the method (level of theory), which inputs and outputs filenames are named after. The g09.sh file appears as #!/bin/bash #PBS -l select=1:ncpus=4 #PBS -l walltime=1:00:00 #PBS -e myjob.err #PBS -o myjob.out #PBS -A train_ccecam17 #PBS -q R #PBS -W group_list=train_ccecam17 cd $PBS_O_WORKDIR module load g09. $g09root/g09/bsd/g09.profile # for bash script # source $g09root/g09/bsd/g09.login # for csh script export GAUSS_SCRDIR=$CINECA_SCRATCH/g09_$PBS_JOBID # def. tmp folder in $CINECA_SCRATCH mkdir -p $GAUSS_SCRDIR # the dir must exist g09 < geometry_mp2_1.com > geometry_mp2_1.log g09 < geometry_mp2_2.com > geometry_mp2_2.log g09 < geometry_mp2_3.com > geometry_mp2_3.log g09 < geometry_mp2_4.com > geometry_mp2_4.log g09 < geometry_mp2_5.com > geometry_mp2_5.log g09 < geometry_mp2_6.com > geometry_mp2_6.log rmdir $GAUSS_SCRDIR # remove tmp folder (works only if empty) this script contains the needed commands to submit the calculation using the com input and sending the output in a log file. The g09.sh script need to be present in each directory from which you send the calculations (each directory will contain the input files.com and the corresponding g09.sh script).

4 7. The calculation can be sent on the PBS queue for execution, with the command qsub g09.sh which should be runned for each folder containing your g09 inputs. To check if the calculations are running use the command qstat qstat u yourcinecausername if a R appears close to you job this is running, if a Q appears the job is still queuing For more information have a look to this web page: - UG2.5.1:BatchSchedulerPBS-RunningapplicationsusingPBS In the meanwhile, (while the calculations run.) B. Exploring optimized geometries produced with different methods We have already optimized the geometry of the parallel displaced benzene dimer using different level of theories. We started the geometry optimization exactly from the same coordinates and we performed geometry optimization at different levels of theory. We will directly provide you the resulting optimization outputs, so that you could directly see how the geometry is changing to minimize the energy (i.e. the geometry optimization procedure). 1. Locally (on the local sissa machine), start Gaussview. i.e. Type: gaussview 2. go to the /scratch/qm/_school/benzene_dimer/minimized_geometries Here you will find the output files and for your curiosity the input files we have used for each optimization. You will find one directory for each method (b3lyp, B3lyp-d3, B2plyp, mp2). In addition you will find also geometry optimizations with the pure DFT (non-hybrid) BLYP functional (blyp) and the Hartree- Fock (HF) method (hf). 3. Visualize the output and follow the geometry changes with Gaussview. What are the main differences that you see in the optimized geometries (with respect to the input ones) of the benzene dimer obtained at the different levels of theory? Once the calculations are finished A. Perform a scan along the intramolecular distance of the benzene dimer - part 2 8. Once the calculations are finished, you can compute the binding energy profiles, as a function of the intermolecular distance using the formula in the Introduction. The electronic energies E dimer (in Hartree units) we need for the B3LYP and B3LYP-d3 methods can be found in the log files in correspondence to the SCF Done string. Analogously, E momer can be found in the monomer log files. For the MP2 method, the SCF Done string will report only the SCF energy. After the SCF, Gaussian computes the correlation energy with the perturbation theory. The second order correlation correction to the SCF energy is printed as E2 =. In the same line, the total MP2 energy (SCF plus E2) is printed as EUMP2. (Optional) The B2PLYP method works in a quite similar way as the MP2 method, so that the SCF Done string will report only SCF energy, the second order correction to the SCF energy is printed as E2(B2PLYP) and in the same line the total B2PLYP energy (SCF plus E2) is printed as E(B2PLYP).

5 The binding energies are usually given in kcal/mol in the most common chemistry literature. In order to convert from Hartree to kcal/mol just multiply by Create an ascii text files, by listing on the first column the relative displacement from the equilibrium structure of the first monomers (i.e. the z coordinate, being z=0 the equilibrium dimer structure) and in the second column the binding energy of the dimer at each point. For each method, you have used add an additional column in the file. You can collect this data on the local SISSA machine. 10. Visualize the output with any graphical program, like xmgrace package: Type on your local window xmgrace nxy filename 11. Looking at the graph try to rationalize the differences also with respect to the geometry optimizations you have visualized before. OPTIONALS: If there is time left or if you feel particularly confident and fast in the procedure ;) 11 [OPTIONAL 1]. You could decide to improve the scan by increasing the density (i.e. making an intermediate steps) making the curve smoother, or by extending the overall scanned distance. 12 [OPTIONAL 2]. You could decide to similarly investigate the parallel-displacement coordinate (according to the input geometry, the y coordinate). This would produce another sets of coordinates and an equivalent series of calculations describing the energy profile (PES) along this other coordinate. References [1] Pavel Hobza, Heinrich L. Selzle, Edward W. Schlag Structure and Properties of Benzene-Containing Molecular Clusters: Nonempirical ab Initio Calculations and Experiments, Chem. Rev., 1994, 94 (7), pp [2] Non-Covalent Interactions : Theory and Experiment, Chapter 3: Potential-Energy and Free-Energy Surfaces Pavel Hobza, Klaus Müller-Dethlefs; print publication date: 18 Nov 2009 [3] S. Grimme, J. Antony, S. Ehrlich and H. Krieg, A consistent and accurate ab initio parameterization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys., 132 (2010) [4] S. Simon, M. Duran, and J. J. Dannenberg, How does basis set superposition error change the potential surfaces for hydrogen bonded dimers?, J. Chem. Phys., 105 (1996)