Molecular Specification

Anan Wu2014-10-10

Typical Gaussian Input

Molecular specification

This input section mainly specifies the nuclear positions. There are several ways In which the nuclear configuration can be specified: as a Z-matrix, as Cartesian Coordinates, or as a mixture of the two.

Gaussian Input file in Cartesian coordinates and in Z-matrix

Cartesian Z-matrix

Z-matrix: is a way to represent a system build of atoms. A Z-matrix is also known as internal coordinate representation. It provides a description of each molecule in terms of bond length, bond angle and dihedral angel, the so-called internal coordinates.

9 variables 3 variables (2 effective)

• What’s the difference between Cartesian coordinates and Z-matrix

• Why one needs to specify the nuclear configuration in Z-matrix? • How to?

Degrees of freedom A degree of freedom of a physical system refer to a real parameter that is

necessary to characterize the state of a physical system

In 3-D space, one can describe the n-particles system with 3N Cartesian coordinates. However, are all these Cartesian coordinates chemically relevant?

Degrees of freedom

𝑅=√ (𝑥1−𝑥2 )2+( 𝑦1− 𝑦 2 )2+ (𝑧 1− 𝑧 2 )2

Let’s say the C atom has coordinate (x1,y1,z1)and the O atom has coordinate (x2,y2,z2) with z2 unknown. Application of the formula for distance between two atoms results in one equation with one unknown, in which we can solve for z2.

R describes the internal motion between two atoms. Hence, it’s called internal coordinate.

Degrees of freedom

3N = 6 = 3 + 2 + 1

Translation Rotation Vibration

Degrees of freedom

Degrees of freedomMonatomic Linear Molecules Non-Linear Molecules

Translation (x, y and z)

3 3 3

Rotation(x, y and z)

0 2 3

Vibration 0 3N-5 3N-6

Total 3 3N 3N

chemically relevant

We generally use the bond length, bond angle and dihedral angle to describe the internal motions (vibrations) of the molecule. This representation is called Z-matrixrepresentation. A skillful choice of internal coordinates can make the interpretation of results straightforward.

• Why one needs to specify the nuclear configuration in Z-matrix?

Why Z-matrix?

In certain cases, Cartesian coordinates are inappropriate to describe internal motions.

How to construct Z-matrix?

Bond length:

2-1 2-1+1 N-1

Bond angle:

3-1 3-1+1 N-2

Dihedral angle:

4-1 4-1+1 N-3

3N - 6

Examples:

N2O CH3OH

Dummy atom

N2O N2O

Dummy atom

Tasks ：

Group 1 ： How to describe the puckering motion of cyclic molecules?

Group 2 ： How to determine the symmetry of a molecule given the atomic coordinates?