### Introduction

# Introduction

*Migen Halo, 2010-10-13*

## 1. Basics

Briefly summarizing the main concepts:**dynamic electron correlation**energy is the energy due to the instantaneous correlation of electron motions- it represents only a small part of the total energy of a system, BUT
- it is essential to explain phenomena involving
**dispersive forces** - it contributes a large fraction of the cohesive energy in
**ionic**crystals - it provides a substantial part of the strength of
**covalent**chemical bonds in molecules and solids.

- the Hartree-Fock (HF) solution does not take into account electron correlation;
- Density Functional Theory (DFT) considers only
*local*electron correlation in its standard formulations or introduces empirical parameters,

**post-HF**- the MP2 (Møller-Plesset at the second-order) perturbative correction currently implemented uses the HF solution as an uperturbed reference state; this solution is provided by the CRYSTAL program
**local**- local functions (called "Wannier functions, WF'') are adopted instead of
delocalized Bloch functions; this permits the exploitation of the
**short-range**nature (*E*∝*R*) of electron correlation. The main advantage of the local approach is the achievement of a^{-6}*scaling*which is much lower (nearly linear -*N*or*N*) than that of "classical'' MPn methods, whose scaling is typically ≥^{2}*N*. The scaling is the quantity of computational resources, in terms of time and memory, as a function of the irreducible number of atoms^{5}*N*of the system. **non-conducting**- the localization procedure and the MP2 method itself are not suitable for conducting systems;
**crystals**- periodic structures have peculiar features, such as translational symmetry, thus their treatment is rather different from that of molecules.

As far as concerns the theoretical background, refer to the publications and to the supporting material to be found in the CRYSCOR webpage; the more significant among them are listed here below:

*CRYSCOR code general presentation:*

C. Pisani, L. Maschio, S. Casassa, M. Halo, M. Schütz and D. Usvyat, Periodic Local MP2 Method for the Study of Electronic Correlation in Crystals: Theory and Preliminary Applications, J. Comput. Chem.,**29**, 2113 (2008).*Symmetry Adapted Wannier functions:*

S. Casassa, C. Zicovich-Wilson and C. Pisani, Symmetry-adapted Localized Wannier Functions suitable for periodic local correlation methods, Theor. Chem. Acc.,**116**, 726 (2006).*MP2 correction to the density matrix:*

C. Pisani, S. Casassa and L. Maschio, On the Prospective Use of the One-Electron Density Matrix as a Test of the Quality of Post-Hartree-Fock Schemes for Crystals, Z. Phys. Chem.,**220**, 913 (2006)

and

D. Usvyat and M. Schütz, Orbital-unrelaxed Lagrangian density matrices for periodic systems at the local MP2 level, J. Phys.: Conf. Ser., Honorary issue Pisani,**117**, 012027 (2008).

## 2. Running CRYSCOR

As just said, CRYSCOR can be considered the post-HF option of the CRYSTAL program, since all the information at the HF level is provided by CRYSTAL. Therefore, to execute a CRYSCOR job, you first have to run a CRYSTAL calculation.Actually, the CRYSTAL calculations to be run are two:

- The first one (
*crystal*executable) performs a wave function calculation. Geometry and symmetry information, Fock and density matrix, canonical eigenvalues and eigenvectors are stored on fortran unit*fort.9*. - The second one (
*properties*executable) calculates on request a number of quantities of interest. By means of the

keyword, localized Wannier functions (WF) are calculated from the subset of occupied (delocalized) crystalline orbitals and symmetrized (keyword**LOCALI**

). The related information is stored on fortran unit**SYMMWF***fort.80*.

For the use of CRYSTAL please refer to the related Tutorials on the CRYSTAL website;
here only the script usage is reported along with the CRYSCOR one.

The *crystal*
and *properties* executables
are to be run according to the following syntax
(input file names to be used without extension):

**runcry09** crystal_input_file

**runprop09** crystal_input_file crystal_f9_unit

*; it needs 3 arguments:*

**runcryscor09**- the name of the
*cryscor*input file, - the name of the fortran file
*.f9*produced in a preliminary*crystal*calculation, - the name of the fortran file
*.f80*produced in a preliminary*properties*calculation, all to be given without extension.

**runcryscor09**cryscor_input_file crystal_f9_unit properties_f80_unitThe extensions of the input and the output files needed/produced are summarized in table 1.

**Table 1:**Input and output standard file names.

executable | input file | output file |

crystal | .d12 | .out |

.f9 | ||

properties | .d3 | .outp |

.f80 | ||

cryscor | .d4 | .outc |

## 4. General input comments

### a. Basic input

The CRYSCOR input consists of a series of keywords (KW) followed by the respective arguments, to be written in free format. The order of KWs is pratically free.A minimal CRYSCOR input file looks as follows (table 2):

**Table 2:**CRYSCOR input sample.

KNET | CRYSTAL eigenvalue calculation |

12 | |

MEMORY | Memory required (megabytes) |

2000 | |

DOMDEF | Definition of the virtual space |

1 | |

1 3 | |

1 2 3 | |

PAIR | Definition of the occupied space |

6. 10. | |

TOBJ | Tolerances |

0.001 0.001 | |

END | |

END | |

**NEWK**

and **MEMORY**

are the only mandatory cards to run
a CRYSCOR calculation.
Recalling that:

- the MP2 technique involves biexcitations of electron pairs from the occupied to the virtual space;
- these spaces are described in terms of
**localized**functions, which are the*Wannier functions*(WF) for the occupied manifold and the*PAOs*- Projected Atomic Orbitals - for the virtual one,

indicates the distance (in Å) between two Wannier functions; if the WF-WF pair distance results within the chosen value, then corresponding product distributions are calculated, otherwise they are neglected.**PAIR**

The

keyword acts then by "reducing'' the occupied space, which is a basic requirement of the local approach.**PAIR**

defines the number of**DOMDEF***stars*to be included in the*domain*:**star**- a star is the set of atoms at the same distance from a reference atom;
let's focus for instance on LiH:
the H
^{-}ion in the reference cell represents the first star, its 6 Li^{+}first neighbours the second star and its 12 second H^{--}neighbours the third star (in the whole 3 stars with a total of 19 atoms),

(1) is not the only way of defining WFs' domains; for molecular crystals, the KW**DOMDEF**

selects all the atoms belonging to a given molecule.**MOLDOM** **domain**- the domain is the set of PAOs associated to each atom belonging to the selected stars.

These keywords concerning the virtual space act again in the sense of reducing it, which is fundamental to achieve the low-scaling typical of local methods.

is the tolerance that affects the tails of the localized functions WFs and PAOs. A 10**TOBJ**^{-3}value provides reliable results.

, **PAIR**

and **DOMAIN**

are then the fundamental parameters of a
CRYSCOR calculation.
The complete list of cards is reported in the CRYSCOR Manual.
**TOBJ**

### b. When approximations are activated...

The computation of bielectronic integrals represents the most expensive part of the calculation, that is why in CRYSCOR are implemented two very efficient approximated techniques for their estimation, namely Density Fitting (DF) and the Multipolar (MP) technique. The two approximations are complementary, since DF is particularly suited for close-by WF-WF pairs, whereas MP can be safely used only for large distances.- To activate DF the following KWs are to be added in the input:
DFITTING DIRECT PG-VTZ ENDDF

The

keyword is followed by:**DFITTING**- the type of DF activated

(see reference M. Schütz, D. Usvyat, M. Lorenz, C. Pisani, L. Maschio, S. Casassa, M. Halo, Density fitting for correlated calculations in periodic systems, Accurate Condensed-Phase Quantum Chemistry, Series: Computation in Chemistry, 27 (2010).)**DIRECT** - the density fitting basis set (type of basis Valence-Triple-Zeta in this case).

- the type of DF activated
- To activate MP you have to add to the input the keyword

, followed by the multipoles order value, which is usually 4. The multipolar computation is activated automatically for distant pairs, that is, referring to the just showed input file, for pairs between 6 and 10 Å, and it is here always explicitly inserted for didactic purposes.**MULTIPO**MULTIPO 4

## 4. Exercises proposed

In the following exercises are proposed on different types of systems, i.e. LiH as a representative of ionic crystals, H_{2}O polymer to check the relation molecular-periodic, NH_{3} as a representative of molecular crystals, and Argon adsorbed on MgO to show the treatment of adsorption phenomena. The different tutorial sections can be chosen from the menu on the left at the top of this page.

LiH and NH_{3} tutorial sessions are specially designed to teach the user to prepare the input and read the CRYSCOR output.

Furthermore, in NH_{3} are proposed:

- an extensive analysis of the localized and symmetrized Wannier functions as provided by CRYSTAL and
- exercises on the MP2 correction to the HF one-electron density matrix.