Exact Analytical Form of Diatomic Molecular Orbitals

Li, Yunzhi; Li, Chen

doi:10.1021/acsomega.2c01905

Cited by 4 publications

(6 citation statements)

References 42 publications

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“…The energy curves of antibonding states are expected to decrease monotonically with

R

from the traditional point of view, however, we have observed shallow local minima along these curves. We note that this is consistent with our recent findings for 3D

{normalH}_{2}^{+}

[48], which reveals the limitation of the concept of bonding/antibonding.…”

Section: Resultssupporting

confidence: 93%

“…Therefore, one can expect to use fewer parameters to accurately describe the energy and wave function for well‐bonded single‐electron systems. In fact, this is also true for 3D

{normalH}_{2}^{+}

where one can use as few as three parameters to accurately describe the energy curve [48]. As an additional remark, Figure 6 suggests that

F (z)

can be used as an indicator to distinguish typical bonding situations from bond‐breaking scenarios where density functional approximations suffer from intrinsic errors [56].…”

Section: Resultsmentioning

confidence: 99%

“…In making the above ansatz, we essentially assume that the wave function has finite derivative at the nodal positions, which is reasonable because this is a necessary condition for eligible wave functions with finite kinetic energy. In fact, this ansatz has also been used in 3D

{normalH}_{2}^{+}

for which we have clarified the analytical forms of all the excited states [48].…”

Section: Excited States Of 1d H Atommentioning

confidence: 99%

“…As an example, here we compare the computational efficiency between the exact analytical solutions and the standard basis expansion methods employed in mainstream quantum chemistry softwares for real‐world

{normalH}_{2}^{+}

. Our solutions were formulated in spheroidal coordinates but have similar structures as the wave functions of 1D model systems presented in this work, see [48] for details. For the basis expansion method, calculations are performed using the PySCF [58–60] code with the augmented correlation consistent polarized n ‐tuple zeta (aug‐cc‐pV n Z, further abbreviated as AV n Z in later text) ( n = D, T, Q, 5, 6) basis set [49, 61].…”

Section: Additional Comparison For 3d Boldh2+mentioning

confidence: 99%

“…The critical key lies in the analytical structure of the exact wave functions. In particular, we have recently proposed a new way of solving single‐electron SEs that can lead us to the exact analytical expression of ground state wave functions that are previously regarded as too complicated to be analytically solvable, such as model problems of one dimensional (1D) hydrogenic atoms and

{normalH}_{2}^{+}

with soft Coulomb potential [46, 47], and real‐world problem of 3D

{normalH}_{2}^{+}

with hard Coulomb potential [48]. Our exact ground state wave functions are cast in a factorized form involving a power prefactor, an exponentially decaying term and a modulator function on the exponential.…”

Section: Introductionmentioning

confidence: 99%

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We extend our previously developed approach for finding the exact analytical form of ground state wave function of single‐electron Schrödinger equations to target arbitrary excited states of 1D model chemical systems of hydrogen atom and normalH2+ with soft Coulomb potential. Our representation of the wave functions is formally unified by a factorized formula involving a polynomial prefactor encoding the nodal information, a non‐integer power prefactor, and an exponentially decaying term modulated by a modulator function. We show that our method leads to faster energy convergence than the basis expansion method, and hence is a better way of representing the wave function. This conclusion is also validated in the real‐world H2+.

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“…The energy curves of antibonding states are expected to decrease monotonically with

R

from the traditional point of view, however, we have observed shallow local minima along these curves. We note that this is consistent with our recent findings for 3D

{normalH}_{2}^{+}

[48], which reveals the limitation of the concept of bonding/antibonding.…”

Section: Resultssupporting

confidence: 93%

“…Therefore, one can expect to use fewer parameters to accurately describe the energy and wave function for well‐bonded single‐electron systems. In fact, this is also true for 3D

{normalH}_{2}^{+}

where one can use as few as three parameters to accurately describe the energy curve [48]. As an additional remark, Figure 6 suggests that

F (z)

can be used as an indicator to distinguish typical bonding situations from bond‐breaking scenarios where density functional approximations suffer from intrinsic errors [56].…”

Section: Resultsmentioning

confidence: 99%

{normalH}_{2}^{+}

for which we have clarified the analytical forms of all the excited states [48].…”

Section: Excited States Of 1d H Atommentioning

confidence: 99%

{normalH}_{2}^{+}

Section: Additional Comparison For 3d Boldh2+mentioning

confidence: 99%

{normalH}_{2}^{+}

with soft Coulomb potential [46, 47], and real‐world problem of 3D

{normalH}_{2}^{+}

Section: Introductionmentioning

confidence: 99%

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Flame Retardant Polymer Nanocomposites: An Overview

Kumar

Dhawan

Rachna

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The overwhelming use of polymer as substitutes for metal, glass, wood, and ceramic has posed the challenge to develop polymer with flame retardant behavior after chemical modification and making composites for effective use of polymer in a wide range of applications. In this context, the present review article describes the advances in the preparation of flame retardant polymer nanocomposite (FPNC) along with the mechanism and properties. The evolved properties of FPNCs are discussed along with illustration and applications such as coating, sheet, and films for different applications in the light chemical composition, size confinement, and dimension alignments.

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Formally Exact and Practically Useful Analytic Solution of Harmonium

Yao,

Yin,

2024

ACS Omega

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We provide a novel exact analytic solution of harmonium with arbitrary Coulomb interaction strength, for ground as well as all the excited states, using our recently developed method for solving Schrodinger equations. By comparing three formally exact analytic representations of the wave function including the one that utilizes biconfluent Heun function, we find that the best and practically useful representation for the ground state is given by an exact factorized form involving a noninteger power pre-exponential factor, an exponentially decaying term and a modulator function. For excited states, additional factors are needed to account for the nodal information. We show that our method is far more efficient than basis-expansion-based methods in representing the wave function. With the exact wave functions, we have also analyzed the evolution trends of the electron density and natural occupation numbers with increasing interaction strength, which gives insight into the interesting physics in the strong correlation limit.

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Exact Analytical Form of Diatomic Molecular Orbitals

Cited by 4 publications

References 42 publications

Is there a better way of representing stationary wave functions than basis expansion?

Is there a better way of representing stationary wave functions than basis expansion?

Flame Retardant Polymer Nanocomposites: An Overview

Formally Exact and Practically Useful Analytic Solution of Harmonium

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