Molecular Structure

Abstract

The first two sections of this chapter define electron probability density and electrostatic potential and illustrate these topics with the water molecule. The ways of apportioning electrons of a molecule to different regions of space are performed by population analysis methods. We illustrate the Mulliken population analysis method for the methane molecule and the natural bond orbitals method for methane, ethene , and ethyne molecules. Next, we present a typical potential energy surface with one first-order saddle point obtained using a combination of Morse functions and calculate intrinsic reaction coordinates for the isomerization reaction HCN → CNH and the symmetric reaction Cl⁻ + H₃CCl → ClCH₃ + Cl⁻. Potential energy profiles for the rotations around the C–C bonds of ethane and 1,2-dichloroethane are presented and discussed. In particular, the potential energy profiles for the staggered conformation of ethane and the synclinal and antiperiplanar conformations of 1,2-dichloroethane, combined with vibrational calculations, enable one to estimate the amplitudes of the corresponding torsional motions. The last section of this chapter comprises chiral molecules, the Cahn–Ingold–Prelog rules for distinguishing R and S enantiomers of carvone, the polarimeter, and the way optically active molecules interact with plane-polarized electromagnetic radiation. At the end of this chapter, the student can find several Mathematica codes (Natural Bond Orbitals for CH₄, Potential Energy Surface, Right and Left Helices, Optical Rotation) with detailed explanations of new commands, a glossary of important scientific terms, and a list of exercises, whose complete answers are given after the appendix.

Appendices

Mathematica Codes

5.1.1 M1. Natural Bond Orbitals for CH₄

Using the results of a Gaussian 09 HF/STO-3G calculation on CH₄, in particular the coefficient matrix that presents the five natural orbitals of CH₄ in terms of the 1s carbon orbital, four sp ³ hybrid atomic orbitals on carbon, and four 1s hydrogen atom orbitals, this Mathematica code shows that the eigenvalues obtained by diagonalizing the density matrix are 2 for the occupied orbitals (the maximum occupation number in accord with the Pauli exclusion principle ) and 0 for the unoccupied orbitals, and the sum of the eigenvalues is equal to the total number of electrons of the molecule. The density matrix is obtained from the coefficient matrix using (5.22).

The Mathematica command Chop replaces real numbers whose absolute values are smaller than 10⁻⁴ by 0. The command Transpose transposes a matrix, and the commands Eigenvalues and Eigenvectors give the eigenvalues and eigenvectors of a square matrix.

5.1.2 M2. Potential Energy Surface

This potential energy surface is modeled for a linear triatomic system XHH as the sum of three Morse functions [see (3.68)], with one of them representing an interaction term (see Baggott et al. 1988). The channels linked by a first-order saddle point correspond to the chemical reaction X + H₂ → XH + H. The dissociation energies for the reactant and product channels are equal to 20 and 50 arbitrary units of energy, and the force constants for the H₂ and XH diatomics are equal to 40 and 80 arbitrary units of energy per square length. The equilibrium point for each Morse function is defined at 1.2.

5.1.3 M3. Right- and Left-Handed Helices

The clockwise screwing motion of a right-handed helix moves the helix away from the observer. From the mathematical point of view, a right-handed helix can be obtained by combining a circle drawn in the clockwise direction with a perpendicular line segment. In a similar way, a left-handed helix can be seen as the result of combining a circle drawn in the counterclockwise direction with a perpendicular line segment. The next two Mathematica codes illustrate the mathematical ways of forming right- and left-handed helices. They use the ParametricPlot3D command, which produces a three-dimensional curve in which the x, y, and z component functions depend on a single parameter (an external variable) that can be interpreted as the time variable:

The following code uses the Mathematica command Manipulate to continuously vary the parameter range, thus visualizing the formation of the helix in the upward direction:

Suggestion: Write a Mathematica code to form right- and left-handed helices that are mirror images of each other.

5.1.4 M4. Optical Rotation

This Mathematica code simulates a wave approaching the observer and gradually rotating its plane of polarization as it passes through an optically active medium. The wave is built by adding left- and right-circularly polarized waves with slightly different phases:

Suggestion: Change the code so that the incoming wave rotates to the counterclockwise direction.

Glossary

Cahn–Ingold–Prelog rules :

Set of rules named after organic chemists S. Cahn (1899–1981), C.K. Ingold (1893–1970), and V. Prelog (1906–1998) that assign different priorities to the atoms or groups bonded to a chiral center . These priorities follow the decreasing order of the atomic numbers, and the atom or group with lowest priority points away from the observer. If the decreasing priority of the atoms or groups bonded to the chiral center and closer to the observer is in the clockwise direction, the chiral center is assigned an R descriptor, whereas for the counterclockwise direction, the chiral center is assigned an S descriptor

Conformers :

Isomers that can be interconverted by internal rotation around a single bond

Electron probability density :

Probability of finding any of the n electrons of a molecule in volume element dv, for any position of the remaining electrons and any spin of all electrons

Electrostatic potential :

Work done by an electric field to transport a test charge (an infinitesimal positive charge) from infinity to a specified point in a charge distribution divided by the test charge. The electrostatic potential for a molecule is usually mapped over an isodensity surface with a specified isovalue (for example, 0.00040) using rainbow colors, where dark blue is used when the electrostatic potential reaches the extreme positive value (electrophilic region ), and dark red indicates the extreme negative value of the electrostatic potential (nucleophilic region )

Enantiomers :

Chiral molecules that are mirror images of each other

Intrinsic reaction coordinate :

Elementary reaction coordinate that starts at the saddle point and progresses down the hill toward the reactant channel (negative values of the reaction coordinate) and toward the product channel (positive values of the reaction coordinate)

Molecular geometry optimization :

Calculation for determining the geometry of a molecule that consists in systematically varying the geometric variables (bond lengths, bond angles, dihedral angles ) until an energy minimum is reached. This type of molecular calculation includes two steps, the first leading to a stationary point (a minimum or a maximum), the second step being a frequency calculation to verify whether the stationary point corresponds to a minimum. If first derivatives increase in the vicinity of the stationary point (positive second-order derivatives), the stationary point corresponds to an energy minimum, and then the frequency calculation will not produce any imaginary vibrational frequency (in Gaussian, imaginary frequencies are presented as negative values)

Mulliken population analysis method :

Method developed by Mulliken for apportioning the total electron charge of a molecule to its atoms and overlap regions

Natural bond orbital method :

Population analysis method that apportions the electron charge of a molecule to Lewis-like electron pairs localized in chemical bonds and atoms

Natural orbital :

Molecular orbital localized in a bond or an atom with maximum electron occupation allowed by the Pauli exclusion principle (number of electrons = 2)

Optical activity :

Ability of a chiral molecule to rotate the plane of polarization of light

Polarimeter :

Instrument for determining the effect of a substance in rotating the plane of polarization of light. A polarimeter essentially consists of a light source, a polarizer, a sample tube, and a polarization analyzer

Population analysis :

Multiple ways of apportioning the total electron charge of a molecule to different regions of space in the molecule, such as atoms and overlap regions (Mulliken population analysis method) or atoms and bonding regions (natural bond orbitals method)

Potential energy surface :

A surface that gives the potential energy of a system of three atoms A, B, and C as a function of the internuclear distances A–B and B–C, with the bond angle $\angle ABC$ being kept constant. A typical potential energy surface shows two valleys (the reactant channel and the product channel) approximately parallel to the corresponding Cartesian coordinate axes and an upper pass between them through one first-order saddle point (see Fig. 5.14). The minimum energy path converts reactants into products, passes through the first-order saddle point , and defines the reaction coordinate

Exercises

• E1. Explain the expression for the electrostatic potential of a molecule.

• E2. Consider a calculation for the H₂ molecule with a minimal basis set and find the coefficients of the 1s orbitals in terms of the overlap integral between these orbitals.

• E3. Consider the HF/STO-3G calculation on the CH₄ molecule. Starting with the coefficient matrix , use Mathematica to obtain the corresponding density matrix .

• E4. Consider the density matrix and the full Mulliken population matrix of the Gaussian HF/STO-3G calculation on the CH₄ molecule. Find the overlap integral between the 1s ₂ and 1s ₃ hydrogen orbitals.

• E5. Consider the full Mulliken population matrix of the Gaussian HF/STO-3G calculation on the CH₄ molecule. Find the populations condensed to the carbon atom and to the overlap region between the carbon atom and hydrogen atom 1 (1s ₁ orbital).

• E6. Derive the value for the tetrahedral angle.

• E7. Write expressions for the sp ³ hybrid orbitals centered on atom C of CH₄ and show that these hybrids are orthonormal.

• E8. Use symmetry and Mathematica to write the expressions for the sp ² hybrid orbitals centered on atom B of BF₃ and show that these hybrids are orthonormal.

• E9. Write expressions for the sp hybrid orbitals centered on atom C of CO₂ and show that these hybrids are orthonormal.

• E10. Derive (5.29).

• E11. The analysis of the vibrational spectra of 1,2-dichloroethane shows that the population of the most abundant antiperiplanar conformer is about 80 % at ambient temperature (El Youssoufi et al. 1998). Assuming that only conformers 1 and 2 exist, calculate the energy difference between these conformers at ambient temperature.

• E12. The molecular formula for tartaric acid is HOOC–CH(OH)–CH(OH)–COOH. Identify and classify the stereoisomers for this molecule.