# Interatomic forces in an ensemble via vector analysis

• ewan_71
In summary, The user has a question about calculating forces between atoms in an ensemble and provides a print out from a quantum mechanical code for reference. They also provide equations for calculating the force on a particular atom in the direction of another and mention that the inter-atomic force can be approximated by adding the forces in both directions if the atoms are only bound to each other.
ewan_71
Hi all,

My question is about obtaining the force between two particular atoms in an ensemble. Please excuse my ignorance on this topic. I can calculate all the forces in the system by the Hellmann–Feynman theorem using any quantum mechanical code. This is the print out from such a code:

Forces on atoms are in units of Ry/au
atom 1 force = 0.00166685 -0.00270340 -0.00251785
atom 2 force = -0.00060500 -0.00337831 0.00397175
atom 3 force = 0.00306646 -0.00125532 -0.00156902
atom 4 force = -0.00081405 -0.00165894 0.00224862

The atomic positions are, in fractional coordinates:
1 0.494721110 0.619850102 0.464420445
2 0.296515967 0.455429577 0.485041062
3 0.263967419 0.580233832 0.449392069
4 0.490069979 0.402504092 0.458079571

As I understand it, the force on atom 1 in the direction of atom 2 is given by:

F(1,2)= (R(1,2)dotF1)*R(1,2),

Where the directional vector is R(1,2) = P2-P1/|P2-P1| and P1 and P2 are the positions of the atoms 1 and 2.

The above gives:

F(1,2)= -0.001278802 0.00172051 -0.0002 and |F(1,2)| = 0.25835556 which is repulsive

Furthermore, how about the force on atom two in the direction of atom one? By the same approach this force is 0.002931211 which is also repulsive.

Is it true then to think that, the 1-2 inter-atomic force is, given by F(1,2)+F(2,1)??

Any help on this topic would be greatly appreciated! Thank you very much in advance!

Ewan

ewan_71 said:
Is it true then to think that, the 1-2 inter-atomic force is, given by F(1,2)+F(2,1)??

As an approximation yes, if they're only bound to each other. If either has bonds to other atoms then you've got a more complicated situation.

,

Thank you for your question. Interatomic forces in an ensemble can indeed be obtained through vector analysis, as shown in your calculation using the Hellmann-Feynman theorem. The forces between two atoms in a system are determined by the positions and orientations of the atoms, as well as the interactions between their respective electronic structures.

In your example, the forces between atoms 1 and 2 are repulsive in both directions, which is expected since the atoms are close to each other and have similar electronic structures. The total force between these two atoms can be calculated by adding the individual forces, as you have suggested. However, it is important to note that this is not always the case for all atom pairs in the ensemble. Some may have attractive forces, while others may have no force at all. It depends on the specific electronic and structural properties of the atoms involved.

I hope this helps clarify your understanding of interatomic forces in an ensemble. If you have any further questions, please don't hesitate to ask. Keep up the good work with your quantum mechanical calculations!

## 1. What are interatomic forces?

Interatomic forces are the attractive and repulsive forces that exist between atoms in a molecule or solid. They are responsible for the physical and chemical properties of materials and play a crucial role in determining the structure and behavior of matter.

## 2. How do interatomic forces affect an ensemble?

Interatomic forces affect the behavior of an ensemble by determining the arrangement and interactions of the atoms within it. These forces can influence the properties of the ensemble, such as its melting point, strength, and conductivity.

## 3. What is vector analysis?

Vector analysis is a mathematical tool used to analyze and understand the forces acting on objects in a physical system. In the context of interatomic forces, vector analysis involves breaking down the forces into their individual components and analyzing their magnitude and direction.

## 4. How does vector analysis help in studying interatomic forces?

Vector analysis is essential in studying interatomic forces because it allows us to understand the complex interactions between atoms and how they contribute to the overall behavior of a material. It helps us visualize and quantify the forces acting on atoms, which is crucial in predicting the properties of materials.

## 5. What are the main types of interatomic forces?

The main types of interatomic forces are ionic, covalent, metallic, and van der Waals forces. Ionic and covalent bonds are strong and directional, while metallic bonds are non-directional and weaker. Van der Waals forces are the weakest and are caused by temporary dipoles between atoms.

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