Comparative Study of Three Methods of Calculating Atomic Charge in a Molecule Wanda Lew Heather Harding Sharam Emami Shungo Miyabe San Francisco State University Tomekia Simeon Jackson State University Source of Wisdom: Sergio Aragon January 16, 2004

Why is assigning charges to various atoms of a molecule of interest? Assigning charge to various atoms allows: Prediction of reactive sites in a molecule Prediction of reactive sites in a molecule Charge distribution determines all molecular properties Charge distribution determines all molecular properties Andrew S. Ichimura SFSU presentation 9/26/03

Why isn’t there just one best method that everyone uses to calculate atomic charge? No concensus on what criteria to use to judge which method is better i.e. No concensus on what criteria to use to judge which method is better i.e. a. Do we arbitrarily say that if a method is basis set independent it is “better”?* b. Or is the better method one that’s able to account for anticipated changes in charge distribution after various perturbations to the molecule such as: ● varying dihedral angles* in a molecule

We Decided to Examine Three Methods for Assigning Charges to Atoms in a Molecule Population Analysis (R.S. Mulliken, 1955) Population Analysis (R.S. Mulliken, 1955) Atoms in Molecule (R.W.F. Bader, 1965) Atoms in Molecule (R.W.F. Bader, 1965) Electrostatic Potential (Merz-Sing-Kollman) Electrostatic Potential (Merz-Sing-Kollman)

What is Population Analysis? This method was proposed by R.S. Mulliken This method was proposed by R.S. Mulliken Sample Molecule: A-B To assign charge on atom A, uses a molecular orbital function represented by a linear combination of the atomic orbitals To assign charge on atom A, uses a molecular orbital function represented by a linear combination of the atomic orbitals   =C A  A + C B  B  =N(C A 2 + 2C A C B S AB + C B 2 ) Mulliken Charge on Atom A would be: Q A =N(C A 2 + C A C B S AB ) Weaknesses: Weaknesses: a. Divides overlap term symmetrically b. Atomic orbital term C A 2 assigned to atom even if the charge on that atom is polarized/diffuse enough to bleed some e- density into neighboring atom

Electrostatic Potential Ability to compute the degree to which a positive or negative test charge is attracted to or repelled by the molecule that is being represented by the multipole expansion. Ability to compute the degree to which a positive or negative test charge is attracted to or repelled by the molecule that is being represented by the multipole expansion. ESP is directly calculated from the electron density using a many electron wavefunction and point charges of the nuclei. ESP is directly calculated from the electron density using a many electron wavefunction and point charges of the nuclei.

Electrostatic potential is both a molecular property and a spatial property. Electrostatic potential is both a molecular property and a spatial property. It depends on what charges exist in the molecule and how they there are distributed. It depends on what charges exist in the molecule and how they there are distributed. The electrostatic potential created by a system of charges at a particular point in space, (x, y, z), is equal to the change in potential energy that occurs when a +1 ion is introduced at this point. The electrostatic potential created by a system of charges at a particular point in space, (x, y, z), is equal to the change in potential energy that occurs when a +1 ion is introduced at this point. It also depends on what point It also depends on what point (x, y, z) we choose to investigate. If we select a point where the +1 charge is attracted by the molecule, the potential will be negative at this point. (x, y, z) we choose to investigate. If we select a point where the +1 charge is attracted by the molecule, the potential will be negative at this point. On the other hand, if we select a point where the +1 charge is repelled, the potential will be positive. On the other hand, if we select a point where the +1 charge is repelled, the potential will be positive.

AIM Let  (r) be the electron density Let  (r) be the electron density Gradient of  (r) is a vector that points in the direction of maximum increase in the density. One makes an infinitesimal step in this direction and then recalculates the gradient to obtain the new direction. By continued repetition of this process, one traces out a trajectory of  (r). Gradient of  (r) is a vector that points in the direction of maximum increase in the density. One makes an infinitesimal step in this direction and then recalculates the gradient to obtain the new direction. By continued repetition of this process, one traces out a trajectory of  (r).

AIM (cont.) A gradient vector map generated for ethene: A gradient vector map generated for ethene: Since the density exhibits a maximum at the position of each nucleus, sets of trajectories terminate at each nucleus. The nuclei are the attractors of the gradient vector field of the electron density. Since the density exhibits a maximum at the position of each nucleus, sets of trajectories terminate at each nucleus. The nuclei are the attractors of the gradient vector field of the electron density.

AIM (cont.) The molecule is disjointly and exhaustively partitioned into basins, a basin being the region of space traversed by the trajectories terminating at a given nucleus or attractor. The molecule is disjointly and exhaustively partitioned into basins, a basin being the region of space traversed by the trajectories terminating at a given nucleus or attractor. An atom is defined as the union of an attractor and its basin An atom is defined as the union of an attractor and its basin

Comparison of 3 Ways to Calculate Charge on Atom in a Molecule (MUL, AIM, ESP) Using 7 Different Molecules a. Molecules Studied: Urea, Proprionitrile, 1,2-difluoroethane, Glycine, Serine, Propylaldehyde, propane, propanol b. Calculation Methods Used: Hartree-Fock (HF) Density Functional (DFT, specifically B3LYP) c. Criteria used to evaluate quality of method: i. independence of basis set (STO-3g, 321g, 631g, 6311g, 6311g*, 6311g**) ii. How charge on atom changes with change in dihedral angles Andrew S. Ichimura SFSU presentation 9/26/03

Basis Set Dependence of MUL, AIM and ESP –HF Method Urea

Basis Set Dependence MUL, AIM and ESP -- DFT Methods Urea

Dihedral Angle Dependence of MUL, AIM and ESP with HF Methods Urea

Dihedral Angle Dependence of MUL, AIM and ESP with DFT Methods

Basis Set Dependence of MUL, AIM and ESP with HF Methods Proprionitrile

Basis Set Dependence of MUL, AIM and ESP with DFT Methods

Dihedral Angle Dependence of Charges on Atoms in Proprionitrile Using MUL, AIM and ESP with HF Methods

Dihedral Angle Dependence of Charges on Atoms in Proprionitrile Using MUL, AIM and ESP with DFT Methods

Glycine

Basis Set Dependence of Charges on Atoms in Glycine Using a Mulliken Population Analysis

Basis Set Dependence of Charges on Atoms in Glycine Using AIM

Basis Set Dependence of Charges on Atoms in Glycine Using ESP

Glycine – Different Dihedral Angles Optimized45º 90º

Dihedral Angle Dependence of Charges on Atoms in Glycine Using MUL with DFT

Dihedral Angle dependence of Charges on Atoms in Glycine Using AIM with DFT

Dihedral Angle dependence of Charges on Atoms in Glycine Using ESP with DFT

Basis Set Dependence of Charges on Atoms in Serine Using MUL, AIM and ESP with Hartree-Fock Methods

Basis Set Dependence of Charges on Atoms in Serine Using MUL, AIM and ESP with Density Functional Theory Methods

Comparison of methods using 6311-G d basis set using DFT and HF

Basis Set Dependence of Charges on Atoms in Propyl Aldehyde Using MUL, AIM and ESP with Hartree-Fock Methods at theta~2.318

Comparison of Mulliken and AIM using HF and DFT methods

Basis Set Dependence of Charges on Atoms in Propyl Aldehyde Using MUL, AIM and ESP with Density Functional Theory Methods

Basis Set Dependence of Charges on Atoms in Propyl Aldehyde Using MUL, AIM and ESP with Hartree-Fock Methods at theta~127.46

Basis Set Dependence of Charges on Atoms in Propyl Aldehyde Using MUL, AIM and ESP with DFT Methods at theta~127.46

Comparison of Charges on Atoms in Propyl Aldehyde Using MUL and AIM as a function rotating carbonyl group

Charges on Atoms in Propyl Aldehyde Using MUL and AIM with HF and DFT Methods as a function of rotating carbonyl group

Comparison of single and double bonded propyl aldehyde!

Comparison of charges using Mulliken and AIM with HF and dihedral angle =

Comparison of Mulliken and AIM for Butyl Aldehyde using HF and dihedral angle ~0.000

Comparison of charge as a function of dihedral angle for butyl aldehyde using HF and DFT with AIM and MUL

Propane Mulliken Charges via HF, Post HF and DFT Methods

Propane Electrostatic Charges via HF, Post HF and DFT Methods

Atoms in Molecules via HF, Post HF and DFT Methods

Conformational Dependence of Charge (Basis Set 6-31gd) H H H H H H H H

Conformational Dependence of Charge (Basis Set 6-311gd)

Propanol Mulliken Charges via HF, Post HF and DFT Methods

Propanol’s Electrostatic Charges via HF, Post HF and DFT Methods

Propanol’s Atoms in Molecules Charges via HF, Post HF and DFT Methods

A Comparsion of Propanol at Varying Dihedral Angles Conformational Dependence of Charge (Basis Set 6-311gd)

C2H4F2C2H4F2C2H4F2C2H4F2

Error 2070 WARNING: RMS ERROR HAS INCREASED. NEWTON STEP FAILED FOR SURFACE SHEET n. WARNING: RMS ERROR HAS INCREASED. NEWTON STEP FAILED FOR SURFACE SHEET n. Many molecules resulted in error 2070 in Gaussian98 when running AIM. (i.e. ethyl formate, alanine, cysteine) Many molecules resulted in error 2070 in Gaussian98 when running AIM. (i.e. ethyl formate, alanine, cysteine)

References: 1. Politzer, P.; Harris, R.R. J. Chem.Phys. 1970, 92, McQuarrie, D.A.; Simon, J.D. Physical Chemistry: A Molecular Approach. University Science Books: Sausalito, California,

Acknowledgments Inspiration for Project: Inspiration for Project: Dr. Sergio Aragon and Dr. Mario Blanco (Their debate about Mulliken vs AIM method assignment of charges on atoms in molecules made this project happen) PASI/Caltech PASI/Caltech