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Kinematic Equations - II

Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant velocity, and the distance traveled due to a change in velocity. Similar to the first kinematic equation, this equation is also only valid when the acceleration is constant throughout the motion of an object.
Suppose a car merges into freeway traffic on a 200 m long ramp. If its initial velocity is 10 m/s and it accelerates at 2 m/s2, then the...

Kinematic Equations - III

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity as a function of the acceleration and distance over which it acts. The fourth kinematic equation does not have an acceleration term and provides the final position of the object at time t in terms of the initial and final velocities. This equation is useful when the value of the constant acceleration is unknown.
Using the kinematic equations,...

Equilibrium Conditions for a Particle

Equilibrium Conditions for a Particle

When an object is in equilibrium, it is either at rest or moving with a constant velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when an object is at rest, while dynamic equilibrium occurs when an object is moving with a constant velocity. In both cases, there must be a balance of forces acting on the object.
To understand the concept of equilibrium, let us first consider the forces acting on an object. When different forces act on an object, they can...

Equations of Equilibrium in Three Dimensions

Equations of Equilibrium in Three Dimensions

When analyzing structures or systems at rest, it is necessary to ensure they are in equilibrium. This is where the vector and scalar equations of equilibrium come into play. These equations are crucial in ensuring a structure is stable and will not collapse or fall apart. The vector and scalar equations of equilibrium provide a framework for analyzing the forces acting on a body.
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...

Equation of Motion: General Plane motion - Problem Solving

Equation of Motion: General Plane motion - Problem Solving

Consider a lawn roller with a mass of 100 kg, a radius of 0.2 meters, and a radius of gyration of 0.15 meters. A force of 200 N is applied to this roller, angled at 60 degrees from the horizontal plane. What will be the angular acceleration of the lawn roller?
The friction between the roller and the ground is characterized by two coefficients. The static friction coefficient is 0.15, while the kinetic friction coefficient is 0.1. These values are crucial in understanding the interaction between...

Equation of Motion: Center of Mass

Equation of Motion: Center of Mass

The equation of motion for a single particle can be expanded to encompass a system of particles consisting of n particles. For any arbitrarily chosen particle within this system, the net force acting upon it is the aggregate of both internal and external forces. Extending this principle to all particles within the system results in the equation of motion for the entire assembly.
Internal forces between any pair of particles manifest as collinear pairs of equal magnitude but opposite directions,...

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Related Experiment Video

Updated: Jul 16, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Higher-order equation-of-motion coupled-cluster methods.

¹William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352, USA. so.hirata@pnl.gov

The Journal of Chemical Physics

|July 21, 2004

Summary

New equation-of-motion coupled-cluster (EOM-CC) methods, including EOM-CCSDTQ, efficiently compute electronic properties for various systems. These advanced methods leverage symmetries to minimize computational costs.

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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Published on: June 8, 2018

Isotopic Effect in Double Proton Transfer Process of Porphycene Investigated by Enhanced QM/MM Method

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Area of Science:

Quantum Chemistry
Computational Physics
Theoretical Chemistry

Background:

Equation-of-motion coupled-cluster (EOM-CC) methods are crucial for calculating electronic excited states.
Existing methods faced limitations in computational cost and symmetry exploitation for higher-order excitations.

Purpose of the Study:

To develop and implement advanced EOM-CC methods (EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ) for accurate electronic property calculations.
To enhance computational efficiency by fully exploiting spin, spatial, and permutation symmetries.
To develop related Lambda equation solvers for coupled-cluster methods up to quadruple excitations.

Main Methods:

Derivation and parallel implementation of EOM-CC methods up to quadruple excitations.
Simultaneous and full exploitation of spin, spatial, and permutation symmetries within spin-orbital formalisms.
Development of Lambda equation solvers for CC methods up to CCSDTQ.
Utilized algebraic and symbolic manipulation programs for automated formula derivation and implementation.
Developed efficient tensor contraction strategies minimizing peak operation cost and exploiting tensor symmetries.

Main Results:

Successfully implemented EOM-CCSD, EOM-CCSDT, and EOM-CCSDTQ methods for computing excitation energies, excited-state dipole moments, and transition moments.
Achieved significant computational savings by leveraging symmetries in tensor contractions, minimizing arithmetic and memory costs.
Developed and validated CC Lambda equation solvers for higher-order CC methods.

Conclusions:

The developed EOM-CC methods and Lambda equation solvers provide a powerful and efficient toolkit for studying electronic properties of closed- and open-shell systems.
The implementation's ability to exploit tensor symmetries leads to substantial reductions in computational resources.
These advancements enable more accurate and feasible calculations of complex molecular systems.