Specifying Molecular Electronic and Geometrical Structures

Specifying Molecular Electronic and Geometrical Structures

Parts I: Cartesian Coordinates

How to Construct Cartesian coordinates of an Atom in a Molecule

Using Cartesian coordinates of H2 as an example

Syntax for the Cartesian coordinate format:Atomic label, x-coordinate, y-coordinate, z- coordinate

O

H1 H2

x

ye.g., C2v H2O

H 0.709 0.554 0.000

0.9104

sp3 hybridization of the O center

Molecular Structure Specification for Water

O

H1 H2

x

yHere is the molecular structure of water that is given in Cartesian coordinates

0 1O 0.000 0.000 0.000H -0.709 0.554 0.000H 0.709 0.554 0.000

Spin multiplicityTotal charge on this molecule

Atomic symbol

x-Coordinate y-Coordinate z-Coordinate

e.g., C2v H2O

(Unit used : angstrom for lengths)

0.9104

The Cartesian coordinate style is not intuitive to give us bond lengths, bond angles, and dihedral angles of atoms of interest!

Parts II: Z-matrix (Internal Coordinates)

Using H4 internal coordinates as an example

H 2 0.9 1 105. 3 120.

e.g., C2 H2O2

1

2

3

4Syntax for The Z-matrix format:Atomic label, Atom 2, Bond length, Atom 3, Bond angle, Atom 4, Dihedral angle

Atomic label: For current atomAtoms 1-3: Previously specified atoms, namely reference atomsBond length: For the bond joining the current atom to atom 2Bond angle: Formed by this bond and the bond joining atom 1, and atom 2Dihedral angle: Formed by the plane containing atoms 2-4 with the plane containing the current atom, atom 2, and atom 1

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format Without Variables

43

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format Without Variables

Here is the molecular structure of hydrogen peroxide in the Z-matrix format

0 1O O 1 1.4H 1 0.9 2 105.H 2 0.9 1 105. 3 120.

Spin multiplicityTotal charge on the molecule

Oxygen atom #2: O2-O1 = 1.4 Ǻ

Oxygen atom #1

e.g., C2 H2O2

Hydrogen #3: H3-O1 = 0.9 Ǻ; H3-O1-O2 = 105

1

2

3

4

Hydrogen #4: H4-O2 = 0.9 Ǻ; H4-O2-O1 = 105; H4-O2-O1-H3 = 120

(Positive dihedral angles correspond to clockwise rotation in Newman projections)

Unit: Angstrom for lengths and degrees for angles

43

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format With Variables

Here is another version of the hydrogen peroxide molecular specification

0 1O O 1 R1H 1 R2 2 AH 2 R2 1 A 3 D Variables: R1 1.4 R2 0.9 A 105. D 120.

Oxygen atom #2: O2-O1 = R1 = 1.4 Ǻ

Oxygen atom #1

e.g., C2 H2O2

Hydrogen #1: H3-O1 = R2 = 0.9 Ǻ; H3-O1-O2 = A = 105

1

2

3

4

Hydrogen #2: H4-O2 = R2 = 0.9 Ǻ; H4-O2-O1 = A = 105; H4-O2-O1-H3 = D = 120

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format With Variables and Constants

Here is the third version of the hydrogen peroxide molecular specification

0 1O O 1 R1H 1 R2 2 AH 2 R2 1 A 3 D Variables: R1 1.4 R2 0.9

Constants: A 105. D 120.

Spin multiplicity

Total charge on the molecule

Oxygen atom #2: O2-O1 = R1 = 1.4 Ǻ

Oxygen atom #1

e.g., C2, H2O2

Hydrogen #1: H3-O1 = R2 = 0.9 Ǻ; H3-O1-O2 = A = 105

1 2

3

4

Hydrogen #2: H4-O2 = R2 = 0.9 Ǻ; H4-O2-O1 = A = 105; H4-O2-O1-H3 = D = 120

Construct a Z-matrix for a More Complex Molecule

The atoms inside the redline all lies in a plane

e.g., Cs, Propene (CH2CHCH3)

Step 1: Specify carbon atoms

0 1C C 1 C2CC 2 C3C 1 A3 Variables: C2C 1.34 C3C 1.52 A3 120.

Charge and multiplicityC: C1 at the end of the double bond

C: C2 on the other end of the double bond

C: C3 Third carbon

A3: C3-C2-C1 bond angle

C3C: C-C single bond length

C2C: C-C double bond length

1.34

1.52120

1.09

1.09


Step 2: Specify the hydrogens on C1 and C2

H 1 H4C 2 A4 3 D4 H 1 H5C 2 A5 3 D5H 2 H6C 1 A6 3 D5Variables:H4C 1.09H5C 1.09H6C 1.09A4 120.A5 120.A6 120.Constants:D4 0.D5 180.

H: H1

H: H2

H: H3

D4: The H1-C1-C2-C3 dihedral

H4C: H1-C1 bond length



D5: The H2(or H3)-C1-C2-C3 dihedral

Note that the decimal points in D4 and D5 must be included !

1.34

1.52120

1.09

1.09

A4: H1-C1-C2 bond angel





Step 3: Specify the planar hydrogen on C3

H 3 H7C 2 A7 1 D5 Variables:H7C 1.09A7 109.5Constants:D5 180.

1.34

1.52120

H: H4


A7: H4-C3-C2 bond angle

D5: D9 = D5, H4-C3-C2-C1 dihedral

1.09

Newman Projections are often used to visualize dihedral angles

1.09



Step 4: Specify the non-planar hydrogens on C3

H 3 H8C 2 A8 1 D8 H 3 H9C 2 A9 1 –D8Variables:H8C 1.09H9C 1.09A8 109.5A9 109.5D8 60.

H: H5

H: H6




D8: D9 = -D8, the H5-C3-C2-C1 dihedral

The geometry of C3 is tetrahedral, and thus the bond angle of each of the hydrogens with respect to the C3-C2 bond is 109.5

Newman Projections are often used to visualize dihedral angles


1.34

1.52120

1.09

1.09



Step 5: List all created internal coordinates in a Z-matrix

1.34

1.52120

1.09

1.09

0 1 C C 1 C2C C 2 C3C 1 A3 H 1 H4C 2 A4 3 D4 H 1 H5C 2 A5 3 D5 H 2 H6C 1 A6 3 D5 H 3 H7C 2 A7 1 D5 H 3 H8C 2 A8 1 D8 H 3 H9C 2 A9 1 -D8Variables: C2C 1.34 C3C 1.52 H4C 1.09 H5C 1.09 H6C 1.09 H7C 1.09 H8C 1.09 H9C 1.09 A3 120. A4 120. A5 120. A6 120. A7 109.5 A8 109.5 A9 109.5 D8 60. Constants: D4 0. D5 180.


Parts III: Mixed Internal and Cartesian Coordinates

e.g., Oh, Cr(CO)6

Specify the complete molecular structure

0 1Cr 0. 0. 0. C 1.93 0. 0. O 3.07 0. 0.C -1.93 0. 0.O -3.07 0. 0.C 0. 1.93 0.O 0. 3.07 0.C 0. -1.93 0.O 0. -3.07 0.C 0. 0. 1.93O 0. 0. 3.07C 0. 0. -1.93O 0. 0. -3.07

Cartesian Coordinates for Cr(CO)6

1.93

1.14

Electron configuration: (Ar)3d54s1

Mixed Cartesian and Internal Coordinates for Cr(CO)5NH3

e.g., Cr(CO5)NH3

1.93

1.14

0 1Cr 0 0. 0. 0. C 0 1.93 0. 0. O 0 3.07 0. 0.C 0 -1.93 0. 0.O 0 -3.07 0. 0.C 0 0. 1.93 0.O 0 0. 3.07 0.C 0 0. -1.93 0.O 0 0. -3.07 0.C 0 0. 0. -1.93O 0 0. 0. -3.07N 0 0. 0. 2.27H 12 HN 1 HNCr 2 0.H 12 HN 1 HNCr 13 DH 12 HN 1 HNCr 13 –DVariablesHN 1.02HNCr 115.D 120.

1.02

2.27

Cartesian coordinates are included in a Z-matrix by specifying the bonded-to atom as 0

Mixed Cartesian and Internal Coordinates for Cr(CO)5NH3

e.g., Cs, Cr(CO5)NH3

1.93

1.14

0 1Cr 0 0. 0. 0. C 0 CCr 0. 0. O 0 3.07 0. 0.C 0 -CCr 0. 0.O 0 -3.07 0. 0.C 0 0. CCr 0.O 0 0. 3.07 0.C 0 0. -CCr 0.O 0 0. -3.07 0.C 0 0. 0. -CCrO 0 0. 0. -3.07N 0 0. 0. 2.27H 12 HN 1 HNCr 2 0.H 12 HN 1 HNCr 13 DH 12 HN 1 HNCr 13 –DVariablesCCr 1.93HN 1.02HNCr 115.D 120.

1.02

2.27

The variable names for Cartesian coordinates are given symbolically in the same manner as for internal coordinates

Parts IV: Using Dummy Atoms in Z-matrices

Use of Dummy Atom X to Fix the Three-fold Axis in C3v Ammonia

e.g., C3v, NH3Cs

:

NX 1 1.H 1 nh 2 hnx H 1 nh 2 hnx 3 120.0 H 1 nh 2 hnx 3 -120.0Variablesnh 1.0hnx 110.0

H2

H3

H1N(X)

NH 1 nhH 1 nh 2 hnxH 1 nh 2 hnx 3 -120.0Variablesnh 1.0hnx 107.5

X

C3v:

The use of dummy atoms within Z-matrices, which are represented by the pseudo atomic symbol X, is to fix a symmetric axis

J. Phys. Chem. B 113 (2009) 5290

X

In the calculations, a dummy atom X is placed in the six-atom cycle

Use of Dummy Atom X to Contact Nonbonding Molecular Fragments

Sources for Geometrical Structural Parameters

Periodic table of elements

Standard references like the CRC series

Published experiments and calculations

hybridization Bond angle Geometry Examples

sp 180 Line CHCH, BeCl2, Hg(NH3)2+

sp2 120 Planar triangle CH2CH2, BF2, [CuCl3]2-

sp3 109.5 Tetrahedron CH4, BF2, [Ni(NH3)4]2+

dsp2 90 Square Ni(CN)42-

dsp3 90, 120 Trigonal bipyramid PCl5, Fe(CO)5

d2sp3 90 Octahedron SF6, Co(CN)6

Hybridization of central ions or atoms

Parts V: Summarize Geometry Specification

1. Cartesian coordinates:

• atomic symbol, x, y, z coordinates of each nucleus

• Gaussian expects values in Angstroms

• convenient because most molecular building programs will output Cartesian coordinates

2. Z-matrix coordinates:

• specify positions of atoms relative to one another using bond lengths, angles and dihedral angles (3N-6 variables)

• Gaussian expects values in Angstroms and degrees

• also called internal coordinates

• one section specifies connectivity, second section specifies values of variables corresponding to bond lengths, etc.

• convenient for PES scans because bonds and angles are defined explicitly

Cartesian & Z-matrix Styles

C1 C2

H3

H4 H6

H5CC 1 B1H 1 B2 2 A1H 1 B2 2 A1 3 D1H 2 B2 1 A1 3 D2H 2 B2 1 A1 5 D3

B4

A3

D2

variables:

B1=1.5B2=1.1A1=120.0D1=0.0D2=0.0D3=180.0

• can simplify by taking advantage of symmetry

• expect C-H bonds to be same lengths use variable B2 for all C-H bonds

• expect H-C-C angles to be the same use variable A1 for all H-C-C angles

• careful, though• assigning the same label to two or more geometric

variables means they have to remain equal throughout entire calculation

1 2 2. 3 1. 4 1.

2 5 1. 6 1. 3456

Geometrical connectivity

Atom1Atom2

Bond orders formed between Atoms 1, 2

Connectivity Specification

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Specifying Molecular Electronic and Geometrical Structures