JP3091035B2 - How to display molecular orbitals - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention treats a molecular orbital as a linear combination of atomic orbitals constituting a molecule, and displays the molecular orbital, in particular, the d-orbital component of the molecular orbital using an image display device. About.

[0002]

2. Description of the Related Art In recent years, with the progress of computers, methods based on computational chemistry, such as the molecular orbital method, have become tools that are indispensable and familiar to general researchers, from tools of some experts. Accordingly, visualization has become indispensable to facilitate interpretation of calculation results. For example, when interpreting the results of the molecular orbital method, it is common to graphically output a charge distribution, an electrostatic potential, a reference vibration of a molecule, and the like to an image display device. Above all, drawing actual molecular orbitals is a way to grasp the image of molecules,
This is an important function in terms of interpreting the reaction mechanism and understanding the binding state.

In order to display such molecular orbitals on an image display device, information on the molecular orbitals is described in a format suitable for the image display device while considering that the display result is easy for a user to understand. There is a need to.

One of the methods for this purpose is to calculate a function value or electron density (existence probability) at each point in space from the calculation results of molecular orbitals, and connect points having a certain function value or density. There is a method of expressing as the shape of the electron cloud. Although this method is excellent from the viewpoint of representing an accurate electron distribution, it requires enormous calculations and is difficult to quickly display, and at the same time, it is difficult to analyze the bonding state, reaction mechanism, etc. There are difficulties.

[0005] In order to make it easy to understand the shape of molecular orbitals and their connectivity, there is another method of displaying the components of molecular orbitals in atomic orbitals for each azimuthal quantum number of each atom. This method of decomposing a molecular orbital into its atomic orbital components and displaying it has been used in the past, such as a “schematic molecular orbital diagram” or “Simple Orbital”.
It is called by the name. By using this molecular orbital diagram, the memory consumption of the computer is reduced quickly and
The molecular orbital can be displayed on an image display device. At that time, the display is performed in a form that depends on the atom, so that it is easy to react, which atom is involved in the molecular orbital,
Etc. can be easily determined.

[0006] For the s orbitals and the p orbitals, the molecular orbitals are conventionally displayed using this schematic diagram of the molecular orbitals. That is, for the s orbital, a coefficient of one function element is obtained for each atom from the calculation of the molecular orbital, and is usually displayed at the atom position by a sphere having a size related to this coefficient. The p orbitals are calculated for each atom as x,
The coefficients of the three function elements y and z are obtained, and are usually displayed in the form of a drop having a size associated with the respective coefficients in the x-axis, y-axis, and z-axis directions of the three-dimensional orthogonal coordinate system.

For example, when drawing a schematic diagram of the molecular orbital of a water molecule, the s orbital of an oxygen atom and the p orbital and the s orbital of two hydrogen atoms are combined and displayed. For example, the s-orbital component is represented by the size of a sphere placed on an atom and its color, and the p-orbital component is represented by a pair of differently colored leaves around the atom and their directions. Can be expressed as a color difference.

[0008] Regarding the s and p orbitals, it is easy to draw a schematic diagram of molecular orbitals in this way, and many molecular modeling software (including personal computer tools) having such a function have been used so far.

[0009]

On the other hand, the d trajectory cannot be simply represented as one vector like the p trajectory. That is, a set of d-orbital components of a molecular orbital is represented by a linear combination of five independent function elements using a spherical harmonic function or the like. And the coefficient of each function element of each atom is
Calculated by molecular orbital method.

If a set of d-orbital components of the molecular orbital are represented by five function elements using a spherical harmonic function, this is obtained by multiplying equation (1) of common coefficients by equations (2a) to (2e). It is represented by a spherical harmonic function.

[0011]

(Equation 1)

When a method based on the conventional schematic diagram of molecular orbitals is used, a set of d-orbitals forms a schematic figure expressing the directional components of these five function elements with a size corresponding to the value of each coefficient. , And at or near the position of each atom. That is, FIG. 2 shows the orbital diagram of each of these function elements. When displaying these as molecular orbitals, these are displayed at or near each atom position on the image display device based on the coefficient of each function element. It is necessary to simultaneously display as a figure of different size.

When the schematic diagrams of many function elements are simultaneously displayed on the image display device, the display result is very difficult for the user to see, and the characteristics such as the orientation of the molecular orbit can be easily determined. Disappears.

On the image display device, it is not easy to create one image based on a plurality of pieces of image information each having a complicated shape while determining the mutual relationship between the pieces of image information. , Low productivity. In other words, for that purpose, a high-performance image display device that can perform high-speed arithmetic processing, has a large data storage capacity, and has a high-definition display, etc., is needed. Cannot be displayed.

The present invention solves the above problem, and displays the calculation result of the d-orbital component of the molecular orbital obtained by the molecular orbital method on an image display device in the form of a schematic diagram of the molecular orbital. An object of the present invention is to provide a method for displaying a molecular orbital in which molecular orbitals can be displayed in such a manner that characteristics can be easily judged, and the productivity is high.

[0016]

The method for displaying molecular orbitals according to the present invention treats molecular orbitals as a linear combination of atomic orbitals constituting a molecule and displays a schematic diagram of molecular orbitals of d-orbital components of molecular orbitals. A method for displaying molecular orbitals at or near each atomic position using an apparatus, wherein a set of molecular orbitals is represented by a linear combination of two function elements in spherical harmonic form or matrix form. And

Here, in order to express a set of d-orbital components of molecular orbitals by a linear combination of two function elements in spherical harmonic form or matrix form, d-orbital components of molecular orbitals are generally calculated by molecular orbital calculation. After expressing as a linear combination of five function elements in the spherical harmonic function form or matrix form that have been used so far, it can be obtained by converting these five function elements.

At this time, the unitary transformation is used for the conversion, and five function elements representing a set of d-orbital components of the molecular orbital are obtained by linear combination, and a set of d-orbital components of the molecular orbital are obtained by linear combination. It can be converted into the two function elements represented.

That is, as described above, a set of d orbits is represented by a spherical harmonic function obtained by multiplying equation (1) of common coefficients by equations (2a) to (2e) as described above. If this is converted into a matrix of the quadratic form, equations (3a) to (3)
e).

[0020]

(Equation 2)

Since the d orbit is a linear combination of the five functions of equations (3a) to (3e), it can be expressed as in equation (4). Here, k _i in Expression (4) is a coefficient of each function element, and i represents each function element.

[0022]

(Equation 3)

Then, for example, d x ² −y ² can be expressed as in equation (5), so that the actual d-orbital function can be expressed as in equation (6). Here, the subscript * represents transposed conjugate.

[0024]

(Equation 4)

When the matrix M representing the characteristics of the d-orbit obtained in this way is diagonalized, the direction and magnitude of the d-orbit can be obtained. That is, the unitary transformation shown in Expression (7) is performed on the matrix M. Where U is a unitary transformation matrix,
E is an eigenvalue matrix, λ is its eigenvalue, and the subscript * represents transposed conjugate. This unitary transformation means rotation of the coordinate system.
That is, the rotation represented by the matrix U is performed on the reference rectangular coordinate system to create a new rectangular coordinate system. As a result, the eigenvalue matrix E becomes the x, y, and z components of the d trajectory in the rotated coordinate system defined by U.

[0026]

(Equation 5)

The fact that the d orbit is represented by two function elements (one when the coefficient of one function element is 0) by the above conversion is shown below by actually developing a relational expression.

Firstly five functions elements ^{d x 2 - y 2, d} xz
, D z ² , d yz, d zx are represented by a ₁ , a ₂ ,
If a ₃ , a ₄ , and a ₅ are used, equation (4) can be transformed into equation (8).

[0029]

(Equation 6)

From the property of the eigenvalue, the determinant of equation (9) is obtained from equation (8).

[0031]

(Equation 7)

By solving equation (9), a cubic equation of equation (10) is derived.

[0033]

(Equation 8)

Assuming that the solution of this cubic equation is α, β, γ, α + β + γ = −B / A = 0, that is, α = − (β + γ) according to the formula of the solution.

Summarizing the above, equation (11) is obtained.
Here, β ′ = β / √3 and γ ′ = γ / √3.

[0036]

(Equation 9)

[0037] That is the set of d orbital component of any molecular orbital, size beta - function element matrix M x ² 'of the function element matrix Mx ² y ² sized gamma' - the sum of z ^2, rotation It will be represented as having been made.

Depending on the molecular orbital method, it may be calculated with a different combination of spherical harmonics than those described here, but essentially the same transformation can be used to achieve the purpose.

When displaying the molecular model on the image display device, the center of each d-orbit obtained in the above procedure is adjusted to the position of each atom constituting the molecule, and the size based on the coefficient of each function element And the image is rotated and defined by the unitary transformation matrix U and displayed. Here, it is preferable that the coefficient of each function element and the size of the graphic on the image display device have a positive correlation. Alternatively, when the image is displayed on the image display device, the image may be displayed as a more simplified graphic that embodies these azimuth components.

[0040]

FIG. 1 shows d-orbits for several ratios of β and γ in a wire frame model by using equation (11). Each of FIG. 1 shows one of the various atoms.
It can be understood that the two d-orbit components are greatly simplified and easy to see as compared with the case where the five orbit diagrams in FIG. In addition, the direction of the orbit is very easy to understand, which is very useful for analyzing the direction of the bond, the reactivity, and the like. When drawing a molecular model on an image display device, the center of one d-orbit of various atoms shown in FIG. 1 is adjusted to each atom position in the molecule, and scaled according to the coefficient of each function element. , And a rotation defined by the unitary transformation matrix U.

The actual hexachlorocopper (II) ion (CuC
l ₆ ) Analyze using the schematic diagram of molecular orbitals in ^4- . FIG. 3 shows the arrangement of the atoms of (CuCl ₆ ) ^4- . This (Cu
For Cl ₆ ) ^4- , the molecular orbitals are first calculated by the extended Huckel molecular orbital method, and each function element for each orbital level and each atom (one s orbital, three p orbitals, and five d orbitals) (A total of eight). Based on this result,
For the molecular orbital at the 27th level, s orbital, p
FIG. 4 shows an example in which the trajectory and the function elements of the d trajectory obtained by using the present invention are displayed on an image display device in a schematic diagram of a combination of a sphere, a cone, and a disk. Although FIG. 4 is a black-and-white figure for the sake of convenience on the paper, it is color-coded and three-dimensionally expressed on an actual image display device, and d orbitals localized on Cu atoms, Cl atoms P localized on
The relationship between the direction and the size of the orbit can be understood very clearly.

[0042]

As described above, according to the present invention, when the calculation result of the d-orbital component of the molecular orbital obtained by the molecular orbital method is displayed on the image display device in the form of a schematic diagram of the molecular orbital, it is not easy for the user. The molecular orbitals can be displayed in a form that is easy to see and the characteristics can be easily judged, and a method for displaying the molecular orbitals with high productivity can be obtained.

That is, the present invention displays the d-orbital component of the molecular orbital by dividing it into atomic orbital components.
By converting the calculation result obtained by the molecular orbital method, a simpler and clearer d-orbit diagram can be drawn. As a result, the shape, directionality, size, and the like of the d-orbit for each atom in the molecule can be accurately and easily understood, and the interpretation of the directionality, reactivity, and the like of the bond is extremely facilitated.

This cannot be obtained by the conventional display method using an electron cloud, and a schematic figure expressing the directional component of a function element is represented by a size based on the value of each coefficient and the size of each atom. By displaying at or near the position,
The feature of the orbital attributed to each atom is clarified, which is extremely useful for analysis of a bonding state, analysis of a chemical reaction, and the like.

[Brief description of the drawings]

FIG. 1 is a display example of various d-orbits according to the present invention.

FIG. 2 shows a conventional example in which a d-orbit is divided into five function elements and displayed.

FIG. 3. Arrangement of atoms in (CuCl ₆ ) ^4-

FIG. ^{4 is} a display example of (CuCl ₆ ) ^4- according to the present invention.

Continuation of the front page (56) References JP-A-1-84389 (JP, A) JP-A-2-301872 (JP, A) New Journal of Chemistry, vol. 13, no. 1, p33-40, R.E. L. Johnston et al. , "The Tens or Surface Harmoni c -Equivalent Orbit al Model of Bondin g in Polyhedral Cl uster Molecules (58) investigated the field (Int.Cl. ^7, DB name) G06F 17/50 JICST file (JOIS)

Claims (3)

(57) [Claims] 1. A molecular orbital in which a molecular orbital is treated as a linear combination of atomic orbitals constituting a molecule, and a molecular orbital diagram of a d-orbital component of the molecular orbital is displayed at or near each atomic position using an image display device. A method for displaying molecular orbitals, wherein a set of molecular orbitals is represented by a linear combination of two function elements in spherical harmonic form or matrix form. 2. A set of d-orbital components of a molecular orbital is represented by a molecular orbital calculation as a linear combination of five function elements of a spherical harmonic form or a matrix form, and then a set of d-orbital components is represented by a linear combination. 2. The method according to claim 1, wherein the five function elements representing the orbital components are converted to represent a set of molecular orbitals by a linear combination of two function elements in a spherical harmonic form or a matrix form. How to display molecular orbitals. 3. Converting five function elements representing a set of d-orbital components by a linear combination into two function elements representing a set of d-orbital components by a linear combination. 3. The method for displaying molecular orbitals according to claim 2, wherein conversion is used.