Helium: energies for singlets and triplets

“Helium: energies for singlets and triplets” discusses rival explanations as to why helium's triplet states lie below corresponding singlets. Computations reported in the literature show that triplet states have more compact wave functions than do corresponding singlet states, even though this makes the electron–electron repulsion energy larger for triplets. A simplified variational method is applied to an attempted calculation of the energies, but is inadequate, showing how helium is numerically unfriendly.

Solving the Non-Relativistic Electronic Schrödinger Equation with Manipulating the Coupling Strength Parameter over the Electron-Electron Coulomb Integrals

The non-relativistic electronic Hamiltonian, H(a)= Hkin+Hne+aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

Solving the Non-Relativistic Electronic Schrödinger Equation with Manipulating the Coupling Strength Parameter over the Electron-Electron Coulomb Integrals

The non-relativistic electronic Hamiltonian, H(a)= Hkin+Hne+aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation

The non-relativistic electronic Hamiltonian, Hkin+Hne+aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the wellknown virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter “a”. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input “a”. The most important findings are, 1, vee(a) is quasi-linear function of “a”, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1) <=> Hne <=> Y0(a=0)) and its consequences in relation to “a”. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund’s rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation

The non-relativistic electronic Hamiltonian, Hkin+Hne+aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the wellknown virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter “a”. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input “a”. The most important findings are, 1, vee(a) is quasi-linear function of “a”, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1) <=> Hne <=> Y0(a=0)) and its consequences in relation to “a”. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund’s rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

Ion channel depolarization increases repulsions between positive S4 charges to drive activation

The positively charged residues, arginine and lysine, of the S4 segments of voltage-sensitive ion channels repel each other with Coulomb forces inversely proportional to the mean channel dielectric permittivity ε. Dipole moments induced at rest potential in the branched sidechains of leucine, isoleucine and valine lend high values of ε to the channel. High ε keeps electrostatic forces small at rest, leaving the channel in a compact conformation closed to ion conduction. On membrane depolarization beyond threshold, the repulsive forces between positive S4 charges increase greatly on a sharp decrease in ε due to the collapse of induced dipoles, causing an expansion of the S4 segments, which drives the channel into activation. Model calculations based on α helical S4 geometry, neglecting the small number of negative charges, provide estimates of electrostatic energy for different values of open-channel ε and numbers of positive S4 charges. When theShakerK+channel is depolarized, the repulsion energy in each S4 segment increases from about 0.2 kcal/mol to about 120 kJ/mol (30 kcal/mol). The S4 expansions lengthen and widen the pore domain, expanding the hydrogen bonds of its α helices, thus providing sites for permeant ions. Ion percolation via these sites produces the stochastic ion currents observed in activated channels. The model proposed, Channel Activation by Electrostatic Repulsion (CAbER), explains observed features of voltage-sensitive channel behavior and offers predictions that can be tested by experiment.SIGNIFICANCE STATEMENTScience walks on two legs, experiment and theory. Experiment provides the facts that theory seeks to explain; the predictions of a theoretical model are then tested in the laboratory.Rigid adherence to an inadequate model can lead to stagnation of a field.The way in which a protein molecule straddling a lipid membrane in a nerve or muscle fiber responds to a voltage change by allowing certain ions to cross it is currently modeled by simple devices such as gated pores, screws and paddles. Since molecules and everyday objects are worlds apart, these devices don’t provide productive models of the way a voltage-sensitive ion channel is activated when the voltage across the resting membrane is eliminated in a nerve impulse. A change of paradigm is needed.Like all matter, ion channels obey the laws of physics. One such law says that positive charges repel other positive charges. Since each of these ion channels has four “voltage sensors” studded with positive charges, they store repulsion energy in a membrane poised to conduct an impulse. To see how that stored energy is released in activation, we must turn to condensed-state physics. Recent advances in materials called ferroelectric liquid crystals, with structures resembling those of voltage-sensitive ion channels, provide a bridge between physics and biology. This bridge leads to a new model, Channel Activation by Electrostatic Repulsion,Three amino acids scattered throughout the molecules have side chains split at their ends, which makes them highly sensitive to changing electric fields. The calculations that form the core of this report examine the effect of these branched-chain amino acids on the repulsions between the positive charges in the voltage sensors. The numbers tell us that the voltage sensors expand on activation, popping the ion channel into a porous structure through which specific ions are able to cross the membrane and so carry the nerve impulse along.This model may someday enable us to learn more about diseases caused by mutations in voltage-sensitive ion channels. But for now, the ball is in the court of the experimentalists to test whether the predictions of this model are confirmed in the laboratory.