In this work, I use a stochastic variational method (SVM) to construct an optimized light-matter coupled wave function. In this work I will look for a solution for the non-relativistic QED Hamiltonian of Pauli-Fierz (PF). In a vacuum, highly accurate measurements and theoretical calculations have been developed for atoms and molecules, with the accuracy of the theoretical calculation reaching the level of 1 MHz for the dissociation energy of the H2 molecule28;49.
In this work, I use stochastic variational computation (SVM) to construct optimized wave functions for the light-matter coupled system. In this work, I will demonstrate the use of such methods to calculate the behavior of the H2 ground state as a function of bond length and photon coupling strength, energy convergence comparisons for ECG and DECG approaches, and the behavior of H− ions. Such systems are typically described at the level of the Pauli-Fertz (PF) non-relativistic QED Hamiltonian30;39;56.
Included in this Hamiltonian is the dipole self-interaction term (⃗λ·D)⃗ 2, where ⃗λ is the photon coupling vector and D⃗ is the dipole moment of the system. In this work, I will introduce a modified Gaussian basis so that the dipole self-interaction term can be removed. In the next part, ⃗a will be defined as a 3-dimensional vector and is a single directed vector for a set of particles of the form ⃗n= (n1, . . . , nN) for some x, y, z = n and ⃗r is a three-dimensional vector formed by the set coordinates of one particle.
In this example, we have a correlation between the particle coordinates and a single particle function centered at⃗ci.
Hamiltonian
The matrix elements of the dipole operatorD⃗ are only nonzero between the angular momentum spatial basis functions dhel±1 in 3D orandm±1 in 2D. In the following work, it is assumed that there is only one important photon mode with frequency ω and coupling⃗λ. The formalism can easily be extended for many photonic modes, but this work focuses on the calculation of matrix elements and it suffices to use a single mode.
Dipole self-interaction
Eliminating the dipole self-interaction
In this way, the self-interaction of the aspherical dipole is eliminated, introducing it into the basis functions. In this section, I will present the results of some numerical calculations performed using DECGs as a basis. In this work, the full efficiency of the DECG basis has not been explored since the preliminary calculations were limited to the approach to an Amatrix of the form.
89 Gaussian functions with the coefficients wk and pk (taken from Ref.4) can approximate 1/r with an error of less than 10−8 in the interval [10−9,1]a.u. For the first example, I will consider a 2D system of 2 electrons confined to a harmonic oscillator potential, with the potential term given by. This problem is analytically solvable29 and will be used to compare the ECG (α= 0) and DECG solution. 15), we ensure that there is no coupling to the photons, since the potential is non-spherical because λ̸= 0.
Each ground state is selected by comparing 250 random sets of parameters and choosing the one that minimizes the energy. Note that ECG would approach the exact result even after more optimization and a much larger base size. In this case, it is assumed that there is only one relevant photon mode with frequency ω= 1.5 a.u.
There are infinitely many photons with energy nℏω(n but only the lowest photon states are coupled to the electronic part. These are small probabilities, however, there is a relatively strong coupling between the electrons and the light. The fact that the energy without coupling (only due to the dipole's self-interaction and Coulomb) is -1.67 a.u.
Increasing the coupling further increases the energy of the H2 (e.g. for λ= 3, E=-1.15 a.u.) suggesting strong coupling strength. When the H2 molecule is coupled to light, the energy minimum shifts slightly to shorter distances. There is a clear shift upward as λ increases, which is due to the dipole-self interaction term pushing them higher. the minimum energy point increase withλ: Eb=1.34 a.u. λ= 3 a.u.), where Eb is the difference between the energy of the molecule and twice the energy of the H atom.
H − ion
In this work I have introduced a new variant of ECG basic functions suitable for problems with non-spherical potentials. One problem that arises is that the Coulomb interaction is more complicated than in the conventional ECG case due to the non-spherical integrals occurring in the interaction part. Possible solutions to this include expanding the Coulomb potential in Gaussian elements, which allows the integration to become analytic. or to use numerical integration. In this work it has been shown that using the DECG basis the coupled light matter equations can be solved efficiently even in cases where the coupling and by extension the dipole self-interaction term is large.
This development opens up the possibility to calculate light-matter coupled pair-body systems with high accuracy in cavity QED systems.
Kinetic energy
In principle, Λx, Λy and Λz can be different if the masses of particles depend on the directions. Using analogous results on the left, the overlap with the kinetic energy operator can be given by.
Potential energy
This form allows us to calculate the matrix elements forδ( ˜w⃗r−⃗r) for a general case without using the specific form of the potential, and to calculate the matrix element of the potential by integration over ⃗r. I do this by defining ⃗tas. 56), the matrix element can be expressed as whereB is a 3×3 matrix given by. with the matrix elements of B defined as. The last integral can be calculated again by Eq. 52) Integrating over⃗r should return the overlap, and using Eq. 56) one gets these results immediately.
Electron-photon coupling
Generalized Gaussian Integrals
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