I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
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Is Gravitation A Result Of Asymmetric Coulomb Charge Interactions?Jeffrey Gold
Journal of Undergraduate Research (JUR), University of Utah (1992), Vol. 3, No. 1, pp. 56-61.
Jeffrey F. Gold
Department of Physics, Department of Mathematics
University of Utah
Abstract
Many attempts have been made to equate gravitational forces with manifestations of other phenomena. In these remarks we explore the consequences of formulating gravitational forces as asymmetric Coulomb charge interactions. This is contrary to some established theories, for the model predicts differential accelerations dependent on the elemental composition of the test mass. The
predicted di erentials of acceleration of various elemental masses are compared to those differentials that have been obtained experimentally. Although the model turns out to fail, the construction of this model is a useful intellectual and pedagogical exercise.
I am Grey N. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
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You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
Is Gravitation A Result Of Asymmetric Coulomb Charge Interactions?Jeffrey Gold
Journal of Undergraduate Research (JUR), University of Utah (1992), Vol. 3, No. 1, pp. 56-61.
Jeffrey F. Gold
Department of Physics, Department of Mathematics
University of Utah
Abstract
Many attempts have been made to equate gravitational forces with manifestations of other phenomena. In these remarks we explore the consequences of formulating gravitational forces as asymmetric Coulomb charge interactions. This is contrary to some established theories, for the model predicts differential accelerations dependent on the elemental composition of the test mass. The
predicted di erentials of acceleration of various elemental masses are compared to those differentials that have been obtained experimentally. Although the model turns out to fail, the construction of this model is a useful intellectual and pedagogical exercise.
I am Joshua M. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Michigan State University, UK
I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments .
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
I am Joshua M. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Michigan State University, UK
I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments .
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Ab Initio Protein Structure Prediction is a method to determine the tertiary structure of protein in the absence of experimentally solved structure of a similar/homologous protein. This method builds protein structure guided by energy function.
I had prepared this presentation for an internal project during my masters degree course.
The ppt Sujoy and I made for the Psi Phi ( An Inter School Competition held by our School). Our Topic was Artificial Intelligence.
Credits:
Theme Images from ESET NOD32 (My Antivirus of Choice)
Backgrounds from SwimChick.net (Amazing designs here)
Credits Image from Full Metal Alchemist (One of my favorite Anime).
A free electron model is the simplest way to represent the
electronic structure and properties of metals.
According to this model, the valence electrons of the constituent
atoms of the crystal become conduction electrons and travel
freely throughout the crystal.
The classical theory fails to explain the heat capacity and the
magnetic susceptibility of the conduction electrons. (These are
not failures of the free electron model, but failures of the classical
Maxwell distribution function.)
Condensed matter is so transparent to conduction electrons
The fundamental theory of electromagnetic field is based on Maxwell.pdfinfo309708
The fundamental theory of electromagnetic field is based on Maxwell\'s equations. These
equations govern the electromagnetic fields, E, D, H, and there relations to the source, f and p_v.
In a source-free region, list the Maxwell\'s equations for time-harmonic fields: Given the Phaser
from of the electric field E? For the above given electric field, is B varying with time? Why?
Solution
Maxwell’s equations simplify considerably in the case of harmonic time dependence. Through
the inverse Fourier transform, general solutions of Maxwell’s equation can be built as linear
combinations of single-frequency solutions:† E(r, t)= E(r, )ejt d2 (1) Thus, we assume that all
fields have a time dependence ejt: E(r, t)= E(r)ejt, H(r, t)= H(r)ejt where the phasor amplitudes
E(r), H(r) are complex-valued. Replacing time derivatives by t j, we may rewrite Eq. in the
form:
× E = jB
× H = J + jD
· D =
· B = 0
(Maxwell’s equations) (2) In this book, we will consider the solutions of Eqs. (.2) in three
different contexts: (a) uniform plane waves propagating in dielectrics, conductors, and
birefringent media, (b) guided waves propagating in hollow waveguides, transmission lines, and
optical fibers, and (c) propagating waves generated by antennas and apertures
Next, we review some conventions regarding phasors and time averages. A realvalued sinusoid
has the complex phasor representation: A(t)= |A| cos(t + ) A(t)= Aejt (3) where A = |A|ej. Thus,
we have A(t)= Re A(t) = Re Aejt . The time averages of the quantities A(t) and A(t) over one
period T = 2/ are zero. The time average of the product of two harmonic quantities A(t)= Re Aejt
and B(t)= Re Bejt with phasors A, B is given by A(t)B(t) = 1T T0 A(t)B(t) dt = 12 Re AB] (4) In
particular, the mean-square value is given by: A2(t) = 1T T0 A2(t) dt = 12 Re AA]= 12|A|2 (5)
Some interesting time averages in electromagnetic wave problems are the time averages of the
energy density, the Poynting vector (energy flux), and the ohmic power losses per unit volume.
Using the definition) and the result (.4), we have for these time averages:
w = 1 2 Re 12E · E + 12H · H (energy density) P = 1/ 2 Re E × H (Poynting vector) dPloss dV =
1/ 2 Re Jtot · E (ohmic losses) (6) where Jtot = J + jD is the total current in the right-hand side of
Amp`ere’s law and accounts for both conducting and dielectric losses. The time-averaged
version of Poynting’s theorem is discussed in Problem 1.5. The expression (1.9.6) for the energy
density w was derived under the assumption that both and were constants independent of
frequency. In a dispersive medium, , become functions of frequency. In frequency bands where
(), () are essentially real-valued, that is, where the medium is lossless,that the timeaveraged
energy density generalizes to: w = 1/ 2 Re 1/2 d() d E · E + 1/2 d() d H · H (lossless case) (.7)
The derivation of (.7) is as follows. Starting with Maxwell’s equations (1.1.1) and without
assuming any particular constitutive relations, we obtain:.
The fundamental theory of electromagnetic field is based on Maxwell.pdfRBMADU
The fundamental theory of electromagnetic field is based on Maxwell\'s equations. These
equations govern the electromagnetic fields, E, D, H, and there relations to the source, f and p_v.
In a source-free region, list the Maxwell\'s equations for time-harmonic fields: Given the Phaser
from of the electric field E? For the above given electric field, is B varying with time? Why?
Solution
Maxwell’s equations simplify considerably in the case of harmonic time dependence. Through
the inverse Fourier transform, general solutions of Maxwell’s equation can be built as linear
combinations of single-frequency solutions:† E(r, t)= E(r, )ejt d2 (1) Thus, we assume that all
fields have a time dependence ejt: E(r, t)= E(r)ejt, H(r, t)= H(r)ejt where the phasor amplitudes
E(r), H(r) are complex-valued. Replacing time derivatives by t j, we may rewrite Eq. in the
form:
× E = jB
× H = J + jD
· D =
· B = 0
(Maxwell’s equations) (2) In this book, we will consider the solutions of Eqs. (.2) in three
different contexts: (a) uniform plane waves propagating in dielectrics, conductors, and
birefringent media, (b) guided waves propagating in hollow waveguides, transmission lines, and
optical fibers, and (c) propagating waves generated by antennas and apertures
Next, we review some conventions regarding phasors and time averages. A realvalued sinusoid
has the complex phasor representation: A(t)= |A| cos(t + ) A(t)= Aejt (3) where A = |A|ej. Thus,
we have A(t)= Re A(t) = Re Aejt . The time averages of the quantities A(t) and A(t) over one
period T = 2/ are zero. The time average of the product of two harmonic quantities A(t)= Re Aejt
and B(t)= Re Bejt with phasors A, B is given by A(t)B(t) = 1T T0 A(t)B(t) dt = 12 Re AB] (4) In
particular, the mean-square value is given by: A2(t) = 1T T0 A2(t) dt = 12 Re AA]= 12|A|2 (5)
Some interesting time averages in electromagnetic wave problems are the time averages of the
energy density, the Poynting vector (energy flux), and the ohmic power losses per unit volume.
Using the definition) and the result (.4), we have for these time averages:
w = 1 2 Re 12E · E + 12H · H (energy density) P = 1/ 2 Re E × H (Poynting vector) dPloss dV =
1/ 2 Re Jtot · E (ohmic losses) (6) where Jtot = J + jD is the total current in the right-hand side of
Amp`ere’s law and accounts for both conducting and dielectric losses. The time-averaged
version of Poynting’s theorem is discussed in Problem 1.5. The expression (1.9.6) for the energy
density w was derived under the assumption that both and were constants independent of
frequency. In a dispersive medium, , become functions of frequency. In frequency bands where
(), () are essentially real-valued, that is, where the medium is lossless,that the timeaveraged
energy density generalizes to: w = 1/ 2 Re 1/2 d() d E · E + 1/2 d() d H · H (lossless case) (.7)
The derivation of (.7) is as follows. Starting with Maxwell’s equations (1.1.1) and without
assuming any particular constitutive relations, we obtain:.
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Sergey Seriy - Modern realization of ThomasFermi-Dirac theory
1. Modern ab-initio calculations
based on Thomas-Fermi-Dirac
theory with quantum,
correlation and multi-shells
corrections
SERGEY SERIY, PH.D., EMAIL: SERGGRAY@MAIL.RU
2. A variational technique can be used to derive the Thomas-Fermi equation, and an
extension of this method provides an often-used and simple means of adding corrections
to the statistical model. Thus, we can write the Fermi kinetic energy density of a gas of
free electrons at a temperature of zero degrees absolute in the form:
5
U = c r 3
2/3
f f where c=(3/10)(3π2)f The electrostatic potential energy density is the sum of the electron-nuclear and the
electron-electron terms. We can write this as
E =U +U = - (
v + v
)r
2
Where vn is the potential due to the nucleus of charge Z, ve is the potential due to the electrons,
e
n ep
n
p p
and the factor of 1/2 is included in the electron-electron term to avoid counting each pair of
electrons twice.
With x denoting distance from the nucleus, the total energy of the spherical distribution is given by
e
5
E = ò c - v n
+ v x dx
( r ( r p (Eq.1)
f
3 2
) )4
2
3. The expression for density on the Thomas-Fermi model, ρ=σ0(E’-V)3/2 (Eq.2)
with σ0=(3/5 cf)3/2, is obtained by minimizing Eq.1 subject to the auxiliary condition that
the total number of particles N, remains constant.
The potential energy V, is a function of position in the electron distribution. E’ is the Fermi energy, or
chemical potential, and is constant throughout a given distribution. The Thomas-Fermi equation follows
from Eq.2 and Poisson’s equation.
The tendency for electrons of like spin to stay apart because of exclusion principle is accounted for by
the inclusion in Eq.1 of exchange energy, the volume density of which is given by:
4
U = -c r 3
ex ex, where c=(3/4)(3/π)1/3
ex Minimization of the total energy now leads to the equation
c - c - E -V = f exr r
( ' ) 0
2
5 4
3
3
3
1
3
Which is quadratic in ρ1/3. From this equation we get
( ( ' 2 ) 3 )
0 0 0 r =s t + E -V +t j , where
0 t = (Eq.3)
f
4 2
c
15
ex c
4. Now Poisson equation with the density given by Eq.3 leads to the
Thomas-Fermi-Dirac equation.
In the following two slides we propose additional energy terms to be included in Eq.l.
The incorporation of these terms leads to a simple quantum- and correlation-corrected
TFD equation.
The quantum-correction energy density follows from a slight change in the derivation due to March
and Plaskett.
March and Plaskett have demonstrated that the TF approximation to the sum of one-electron
eigenvalues in a spherically symmetric potential is given by the integral:
I = òò l + E n l dn dl r r 2 (2 1) ( , ) (Eq.4)
where the number of states over which the sum is carried is written as
N = òò l + dn dl r 2 (2 1) (Eq.5)
Here E(nr,l) is the expression for the WKB(quasi-classic) eigenvalues considered as functions of
continuous variables;
nr is the radial quantum number;
l is the orbital quantum number;
and the region of integration is bounded by nr = -1/2, l= -1/2, and E(nr,l)=E’.
5. We have included a factor of two in these equations to account for the spin
degeneracy of the electronic states. The Fermi energy E’ is chosen so that Eq.5 gives the
total number of states being considered, the N electrons occupying the N lowest states.
With considerable manipulation, Eq.4 becomes TF energy equation:
2 3
I = ò (3 P +V P 4
p
x2dx
2
p
3
)
5 2
(Eq.6)
and Eq.5 reveals the TF density through the expression
3
4
3
N = ò P p
x2dx
2
p
(Eq.7)
both integrals being taken between the roots of E’=V(x). We have written these results in atomic
units, so that P, the Fermi momentum, is defined by
P = 2(E'-V)
It is pertinent to examine the error in the TF sum of eigenvalues, as given by Eq.6, for the case of the
pure Coulomb field. The WKB eigenvalues in a Coulomb field are given by
2
E z
, )
2( 1)
(
+ +
= -
n l
r
nr l
6. And let us consider the levels filled from n=1 to n=v, where n is the total quantum
number defined by n=nr+l+1.
Then, for any value of v we can evaluate the error in the TF approximation to the sum of
eigenvalues, comparing always with the correct value, -Z2v. Scott correction to the total
binding energy is obtained by letting v become very large.
Although the sum of one-electron eigenvalues is not the total energy of the statistical atom
because of the electron-electron interaction being counted twice, we might expect to improve
the calculated binding energy greatly by correcting this sum in some manner, since the chief
cause of the discrepancy is certainly the large error in the electron-nuclear potential energy.
This correction can be performed by imposing a new lower limit on l in the integrations above.
When we introduce a new lower limit lmin and a related quantity which we call the “modification
factor”,
1
2
a = lmin +
we obtain, after more manipulation, slightly different expressions corresponding to Eq.6 and Eq.7.
From these revised expressions we can identify a quantum-corrected TF density expression,
3
r =s E -V - a 2
(Eq.9) ,and a corrected kinetic energy density, r r 2
2
0 )
2
2
( '
x
2
U = c 5
3
+ a k f (Eq.10)
2x
7. Revised lower limit on the volume integrals, say x1, is the lower root of E’-V-a2/2x2=0 (Eq.11)
for x<xl, ρ must vanish(stay zero), and we have thus termed x1 the “inner density cutoff
distance”.
We can call the second term on the right-hand side of Eq.10 the “quantum-correction
energy density” and write it in the more consistent form:
c
U q
c a2 q = (Eq.12)
q = ,by defining 2
r x2
The modification factor a, is determined by the initial slope of the potential function.
For interpreting these results it is helpful to consider just what we have done in changing the lower
limit of the orbital quantum number.
Since the lower limit l=-1/2 must correspond to an orbital angular momentum of zero, we have,
clearly, eliminated states with angular momentum of magnitude between zero and a cutoff value
Lc=aħ. Corresponding to Lc at every radial distance is now a linear cutoff momentum:
Pc=aħ/x , and we can rewrite Eq.9 in terms of the Fermi momentum and cutoff momentum:
2
2 2 3
s
c = P - P
0 ( )
2
3
2
r
8. At radial distances less than xl, momenta are prohibited over the entire range from zero
to P, so the electron density vanishes.
This interpretation must be modified somewhat when exchange and correlation effects
are included; for then the Fermi momentum is no longer simply given by Eq.8, except
very near the nucleus.
We can define x1 as in the absence of interactions, i.e., as the lower of the roots of Eq.11, but it is not
correct to demand that the density vanish at the upper root. Instead, we require only that the
density be real.
Correlation Correction. The original TF equation describes a system of independent particles, while
the introduction of exchange energy, which leads to the TED equation, represents a correction for
the correlated motion of electrons of like spin. The remainder of the energy of the electron gas is
termed the correlation energy, by its inclusion we are recognizing that electrons, regardless of spin
orientation, tend to avoid one another.
In extensions of the statistical model there have been suggested at least two different expressions,
for the correlation energy that approach, in the appropriate limits, Wigner’s low-density formula and
the expression due to Gell-Mann and Brueckner at high densities. In addition to these, Gombas and
Tomishima have utilized expansions of the correlation energy per particle in powers of ρ1/3 about the
particle density encountered at the outer boundary of the atom or ion. In this expansion, the term
of first-order can be considered as a correction to the exchange energy, and it follows that the TFD
solutions for a given Z then correspond to correlation-corrected solutions for a modified value of Z.
9. Aside from rather poor approximation of the correlation energy, a drawback to this
procedure is that the TFD solutions must be at hand. If solutions representing specified
degrees of compression are desired, the method would appear to be impractical.
It is interesting and fortunate that over density range of interest it is apparently possible to
approximate the correlation energy per particle quite closely by an expression of form:
6
1 c cr u = -c
Where we have set cc = 0.0842, and compared this approximation with the values due to
Carr and Maradudin.
Derivation. From the results of the preceding slides, we can now express the total energy
per unit volume of the charge distribution in the form:
U c 5 c c v v c )
q
r r 3
r r r 2
4
3
2
6 (
7
x
e
n
f ex c = - - - + +
Where all quantities appearing in the equation have been previously defined. By minimizing the
integral of U over the volume occupied by the charge, while requiring that the total number of
electrons be fixed, we obtain the following equation:
2 r -t r -u r - R = ,where
0
4
6
1
0
3
1
1
3
ex
c
f
4
c
5
u = 7 c s 2
c 4 ( ' ) 2
1 t = 3
0 0 6
2
3
0 x
c
R = s E -V - q
(Eq.14)
10. The electron density is found as a function of R by solving Eq.14, a quartic in ρ1/6.
To accomplish this we write a “resolvent cubic equation” in terms of another variable, say
y:
y3+τ1y2+Ry+(τ1R-υ0
2)=0 . (Eq.15)
Let us use the same symbol y, to denote any real root of this cubic equation.
We can then express the four roots of the quartic, and hence four expressions for the electron
density, in terms of y. One of these expressions possesses the proper behavior in reducing to
previously obtained results in the neglect of correlation and exchange effects, namely:
r = 1 t +y + y 2 + R 3
(Eq.16), where ( 2 2 )
1 ( )
8
1 1 y = t + y t - y + y + R (Eq.17)
We note that ψ vanishes when correlation is neglected, since y=-τ1 is then root of Eq.15. In the familiar
manner we now define a modified TFD potential function θ by the relation: Zθ=(E’-V+τ0
2)x (Eq.18)
and from Poisson equation and Eq.16 we obtain
px t y
ïî
ïí ì <
y R x x
¢¢ = + + + ³
( ) ; ,
2
R Z a
(Eq.20)
2
0 = s q - -t
q (Eq.19), in terms of θ, )
0; .
2
1
1
2 3
1
x x
Z
4 ( 2
2
2 0
3
x
x
11. Eq.20, Eq.15, Eq.17, and Eq.19 constitute the differential relationship to be satisfied at each
step in the integration. We could, of course, write immediately the solutions of Eq.15 in
analytic form, but it proves convenient in the numerical treatment to obtain a root by the
Newton-Raphson method, since a good first guess in the iteration is available from the
previous integration step.
The boundary conditions on Eq.19 are:
First - as the nucleus is approached the potential must become that the nucleus alone, or
θ(0)=1,
Second – at the outer boundary x2, of the distribution of N electrons,
2
= ò = ò ¢¢
N r px dx Z q xdx
1
2
1
4 2
x
x
x
x
Integration by parts yields
N x xx= - ¢ 2
1 (q q ) ,and since ( ) 1 ( ), 1 1 1 q x = + x q ¢ x
Z
we have the usual condition:
(x ) = x ¢(x ) + Z - N 2 2 2 q q (Eq.21)
Z
12. In addition to potential and density distributions, total binding energies of atoms are of
special interest to us here. For the proper evaluation of energies, the arbitrary constant
that is present originally in both the electrostatic potential energy and the Fermi energy
must be specified. The state of infinite separation of the constituent particles is normally
taken to have zero energy.
We therefore follow the usual convention and fix the potential at the edge of the neutral atom at
zero for all values of x2. For an ion the potential energy of an electron at the boundary is taken as:
V = - Z - N
2 x
The defining relation, Eq.18, now gives at the boundary:
or, solving for the Fermi energy,
E Z x
( ) 2 Z N
t
The total electron-nuclear potential energy is given by
E Z r p
While for the electron-electron potential energy we have
2
Zq x = E¢ + Z - N +t
2
0
2 ( ) ( )x
2
x
2
0
2 2
q
¢ = - - -
x
x
2
= -ò
n
p x dx
1
4 2
x
x
x
2
1 x
= ò
e ep
E v r px dx
1
4 2
2
x
13. From Eq.18 and the relation V=-(vn+ve), this becomes:
2
E 1 E t N E N Zq r p
= - + 0 + ¢ - ò
2 2
( 4 )
2
Other energy integrals are, with an obvious notation:
1
x
x
n
p
ep
x dx
x
5 r 4p
E = c ò 3 x 2
dx f f
= ò x dx
x
E cq q
2
2 r 4p
4 r 4p
E = -c ò 3 x 2
dx ex ex
7 r 4p
E = -c ò 6 x 2
dx c c
14. Results. It was pointed out in the introduction that the quantum-corrected
TFD equation yields atomic binding energies in good agreement with
experimental values and with the results of Hartree-Fock calculations:
Z ETFD (a.u.) EHFS (a.u.) Eexp (a.u.)
2 -2.958 -2.878 -2.903
3 -7.601 -7.226 -7.476
4 -14.944 -14.255 -14.665
5 -25.319 -24.079 -24.652
6 -38.995 -37.079 -37.846
7 -56.225 -53.587 -54.589
15. The electron density(external shells) of the rare atoms - helium, neon,
argon, krypton, computed on the present model agree closely with
their crystal radii.
He Ar Ne Kr
16. Thank you for you attention!
Conclusions:
“Keldysh Institute of Applied Mathematic”, Moscow, Russian Federation
“Tohocu University”, Sendai city, Japan