Top page (correct Bohr model including helium. )

Schrodinger equation is equal to Bohr model.

Electron spin is an illusion.

- Pauli exclusion principle, and wavefunction.
- How "stationary" electron wavefunctions can repel each other ?
- Various molecules by quantum chemistry.
- Schrodinger equation is equal to Bohr-Sommerfeld model, again.

*(Fig.1) ↓ Is this useful ?*

As shown on this page, the present quantum chemistry **NEVER**
tries to ask true mechanisms of Pauli exclusion principle and wavefunctions.

So, all we can rely on is Schrodinger equation and very **complicated** mathematical **determinants**.

In determinants like Fig.1, when two orbitals (= matrix lines ) are the **same**, this determinant becomes **zero**, which describes Pauli exclusion principle, they insist.

*(Fig.1') One example -- doubly excited determinants.*

Even in small multi-electron atoms, we **cannot** solve Schrodinger equation exactly.

Dealing with large molecules such as protein interaction by Schrodinger equation is much more **impossible**.

But they just **give up** pursuing true machanism of wavefunction.

As shown on this page and this page, if we leave Pauli exclusion principle as it is (= only **anticommutator** of creation and annihilation operators ), **imaginary** concepts of spin, photon, phonons and virtual particles need to be created.

*(Fig.2) Why quantum chemistry can give almost exact values ?*

Using complicated wavefunction of Fig.1 and **variation** methods, the quantum chemistry can give almost exact values in various atoms and molecules.

The important point is that most textbooks **NEVER** say about **true** reasons why the energies gotten by Schrodinger equation become close to experimental values.

They just say all we do is to **blindly** believe Schrodinger equation, and **don't** ask what the wavefunction really is.

But science should **NOT** be a religion.

In fact, the answer to these questions is these quantum mechanical methods completely rely on Coulomb energy and an integer times **de Broglie** wavelength through **hydrogen**-like wavefunctions ( 1s, 2s, 2p, ... ).

Using these hydrogen's wavefunctions ( with various energy levels ) and **variation** methods, we can get almost exact values in various atoms and molecules. This is the truth.

*(Fig.3) de Broglie wavelength + variation methods = heart of quantum chemistry.*

Hartree-Fock method and configuration interaction in quantum chemistry use variational functions based on **hydrogen** atomic orbitals such as 1s, 2s, 2p, 3s, 3p ...

And each **coefficient** and **charge** Z are usually treated as **variational** parameters.

They find the lowest energies by varying those variational parameters.

The important point is that the concept of an integer multiple of **de Broglie** wavelength is **indispensable**, and **used** in these variational methods **secretely**, though most textbooks **don't** try to reveal this important fact.

As I explain later, hydrogen-like variational functions **always** keep their specific de Broglie wavelegnth while finding the proper charge Z.

*(Eq.1) Schrodinger wavefunctions.*

For example, we use 1s and 2p hydrogen like wavefunctions of Eq.1 to explain the mechanism of variation methods.

This Schrodinger's hydrogen like atom (= nuclear charge **Z** ) gives energy levels of

*(Eq.2)*

All these energy levels are completely **equal** to those of **Bohr model**.

In Bohr model, the principal quantum number **n=1** means this orbit is **1 ×** de Broglie wavelength.

Considering the same energies in them, Schrodinger's wavefunctions **also** obey **an integer** times de Broglie wavelength ?

In fact, as shown in this section, Schrodinger's wavefunctions always **adopt** the same concept of de Broglie wavelength as Bohr-Sommerfeld model !

Though most textbooks **NEVER** say this very important and **true** fact.

*(Eq.3) Virial theorem.*

As shown on this page, Schrodinger's hydrogens always **satisfy** Virial theorem like Bohr model.

In Eq.3, the average potential energy V is **twice** the total energy E.

The total energy E is the sum of potential energy V and the kinetic energy T.

From Eq.2 and Eq.3, the potential energy V is twice the energy E,

*(Eq.4) *

where

*(Eq.5) *

**r _{n}** is the average orbital radius both in Schrodinger's hydrogen and Bohr model.

"

So, when n = 1 and n = 2, each average orbital radius becomes

*(Fig.4) Orbital radius in each energy level.*

When nuclear charge Z = 1 and n = 1, the orbital radius a_{0} is called "Bohr radius", which is important value both in Schrodinger's hydrogen and Bohr model.

*(Eq.6) Variation functions 1s, 2p.*

In quantum chemistry, hydrogen-like wavefucntions including variational parameters Z_{1} are used.

These are called Slater-type orbitals, and variational change Z_{1} is called orbital exponent.

Eq.6 is 1s and 2p Slater-type orbitals.

Using Eq.6, we change parameter Z in wavefunction into Z_{1},

*(Eq.7) *

Here we change **only** variational parameter Z ( → Z_{1} ) included in electron's wavefunction.

And the nuclear charge Z is **not** changed

*(Fig.5) Changing parameter Z _{1} in wavefunction.*

Considering the situation of Fig.5, and substituting Z_{1} into Eq.4 and Eq.5, the orbital radius r_{n} and potential V become

*(Eq.8) *

And when Z ( → Z_{1} ) included in wavefunctions changes, average kinetic energy (= T ) becomes

( using Virial theorum T = -E )

*(Eq.9) Average kinetic energy T.*

In Eq.7, Eq.8, and Eq.9, only charge parameter Z included in the wavefunction itself is changed into Z_{1}.

And the nuclear charge Z **doesn't** change. ( See Fig.6 )

Because charge Z_{1} in the wavefunction is used as variation parameter in quantum chemistry.

As a result, all Z in the kinetic energy T are changed into Z_{1}, because the information of kinetic energy T is included inside wavefunction ( Eq.9 ).

And only one Z in the potential energy V is changed in to Z_{1}, because one of Z in V originates from nuclear charge Z. ( See Eq.8. )

*(Fig.6) Changing variation parameter into Z _{1} inside wavefunction.*

New total energy E' is the sum of potential energy V' (= Eq.8 ) and kinetic energy T' (= Eq.9 ),

*(Eq.10) *

Varying parameter Z_{1} in Eq.10, we find the total energy E becomes the **lowest**, when this parameter Z_{1} is just **equal** to nuclear charge Z, as follows,

*(Eq.11) When Z _{1} = Z → total energy E becomes the lowest.*

This is the **true** mechanism in variational principle used in all ab-initio quantum mechanical methods such as Hartree-Fock and configuration interaction (= CI ).

The result of Eq.11 indicates that variational parameter Z_{1} included in the wavefunction needs to be the **closest** to the **true** charge Z which the electron **feels**.

And the important point is that this mechanism always applies in wavefunctions of **all** energy levels "n". ( See Eq.11. )

*(Fig.7) When variational parameter Z _{1} = Z, the energy E is the lowest and stable.*

Fig.7 is the summary of the mechanism used in variational methods.

For example, if you use 1s hydrogen like wavefunction, this wavefunction is **always** 1 × de Broglie wavelength.

And when the variational charge parameter Z_{1} is just equal to nuclear charge Z, which strength the electron **feels**, the total energy E becomes the lowest and stable.

When this Z_{1} inside wevefunction is **bigger** than Z ( Fig.7 left ), the kinetic energy T based on the wavefunction becomes **too big**. This increased kinetic energy T **overcomes** decreased potential energy V, so the total energy E is **higher**.

When Z_{1} is **smaller** than Z, the atomic radius becomes larger, which leads to **higher** potential energy V.

As a result, when **Z _{1} = Z**, the total energy E is the

*(Fig.8) When Z _{1} = Z → the energy E is the lowest also in n = 2 wavefunction.*

Een if you use **2p** wavefunction ( **n = 2** ), the variational mechanism of Fig.7 **applies**.

As shown in Eq.11, **irrelevant** to the principal quantum number "n", when Z_{1} inside wavefunction is just equal to nuclear charge Z, the total energy E is the lowest.

The main difference between 1s and 2p wavefunctions is 2p fucntion expresses **2 ×** de Broglie wavelength.

So if you use 2p wavefunction as trial function, the atomic radius tends to be **bigger** than 1s wavefunction.

( When you try to make atomic radius smaller by increasing Z_{1}, the total energy E becomes **higher** and unstable. )

*(Fig.9) Angular momemtum L = 0 in S wavefunction → destructive interference of de Broglie wave ! *

Of course, Schrodinger equations completely **depend** on de Broglie theory ( p = h/λ ) in calculating various energies.

In spite of this fact, quantum mechanics **hates** to admit **real** de Broglie wave. Why ?

And as shown in this section and this page, we can prove that Schrodinger's hydrogen **also** satisfies **an integer** number of de Broglie wavelength.

In Bohr-Sommerfeld model, the orbit becomes elliptic or circular, and **cannot** be linear ( L = 0 ).

Because when the angular momentum L is zero like S function, de Broglie waves with **opposite** phases **cancel** each other out.

" An integer times " means the wave phases of these de Broglie waves **fit** each other at their ends, and **don't** cancel.

( And the electron always **collides** with nucleus, when L = 0. This is **impossible**. )

But in Schrodinger's hydrogen including **S** wavefunctions, the angular momentum L **can** be **zero** !

In these cases, the opposite phases of electron's de Broglie waves **cancel** each other out (= destructive interference ), and the electron is **expelled** from this orbit, according to Davisson-Germer interference expeimenrt.

This is one of reasons why they **NEVER** admit **real** de Broglie waves in quantum mechsnics, and **strange** many-world **superposition** becomes dominant in top journals even now.

*(Eq.12) Schrodinger equation for helium atom.*

They insist quantum mechanics can give almost exact energy values in helium atom.

Of course, we **cannot** solve three-body helium atom, so we have to rely on some approximations.

Eq.12 is Hamiltonian (= total energy ) of helium atom.

Helium (= He ) contains two electrons (= 1, 2 ), and Z = 2 positive nucleus.

The first two terms of Eq.12 are kinetic energies of two electrons.

And the remaining terms are potential energy among Z=2 nucleus and two electrons.

r_{12} means the distance between two electrons.

*(Eq.13) 1s × 1s wavefunctions in He.*

They use 1s type variational functions ( see Eq.6 ) in two electrons.

"Z_{1}" is variational charge parameter.
"a_{0}" is Bohr radius.

From wavefunction of Eq.13 and Hamiltonian of Eq.12, total energy W becomes

*(Eq. 14)*

By solving Eq.14, we obtain

*(Eq. 15)*

In Eq.15, E_{1s} is -13.606 eV.

We vary variational parameter Z_{1} to find the lowest energy W.

Differentiating W with respext to Z_{1},

*(Eq.16) *

As shown in Eq.16, when variational parameter Z_{1} is **1.687**, the total energy of this helium becomes the lowest.

*(Eq.17) *

The experimental value of ground state energy of helium is **-79.005 eV**.

Eq.17 is close to this experimental value, but a little different.

Because quantum mechanical stationary wavefunctions **cannot** treat with two electrons **avoiding** each other.

Qunatum mechanis **hates** particle's motion, and it just shows probability density.

For two negative electrons to avoid each other, we have to change stationary wavefunctions into **dynamic** waves.

But this dynamic wavefunctions clearly **contradicts** the basic concept of **probability** density in wavefunction.

As you feel, if we try to describe two electrons **avoiding** each other, we have to treat two electrons as real **moving** particles.

*(Fig.10) Hydrogen and Helium atoms.*

As shown on top page, if we think two orbital planes of helium are **perpendicular** to each other, we can naturally obtain true ground state energies. ( **-79.0037 eV**, the slight difference of -0.0013 eV is relativistic effect, which is good. )

The reason why we have to make two orbits perpendicular to each other is to **avoid** detructive interference of two de Broglie waves.

So considering this desructive interference of de Broglie waves, **Pauli exclusion** principle (= third electron of Li cannot enter 1s orbit ) can be **naturally** explained by this new Bohr model.

**Spin-Spin** interaction is too small (= fine structrue level, **0.00005 eV** ) to overcome Coulomb attractive force, because Pauli exclusion principle is very **strong** repulsive effects (= greater than **10 eV** in Li ).

Atoms | r1 (MM) | WN x 4 | Circular orbit | Result (eV) | Experiment | Error (eV) |
---|---|---|---|---|---|---|

He | 3074.0 | 1.000000 | -83.335 | -79.0037 | -79.0051 | 0.001 |

Li+ | 1944.5 | 1.000000 | -205.78 | -198.984 | -198.093 | -0.89 |

Be2+ | 1422.0 | 1.000000 | -382.66 | -373.470 | -371.615 | -1.85 |

B3+ | 1121.0 | 1.000000 | -613.96 | -602.32 | -599.60 | -2.72 |

C4+ | 925.0 | 1.000000 | -899.67 | -885.6 | -882.1 | -3.50 |

N5+ | 788.0 | 1.000000 | -1239.8 | -1223.3 | -1219.1 | -4.20 |

O6+ | 685.3 | 1.000000 | -1634.38 | -1615.44 | -1610.70 | -4.74 |

F7+ | 607.3 | 1.000000 | -2083.3 | -2062.0 | -2057.0 | -5.00 |

Ne8+ | 544.5 | 1.000000 | -2586.7 | -2563.0 | -2558.0 | -5.00 |

Surprisingly, this new atomic structure of Bohr's helium is applicable to **all other** two and three electron atoms ( ions ).

About the calculation method, see this page.

*(Fig.11) Variation parameter Z _{1} needs to be equal to true charge. *

As I said in Eq.17, when variational parameter Z_{1} is **1.687**, this ground state energy becomes the lowest (= **-77.49 eV** ).

This value of 1.687 is proper, because due to another electron's existence, the central charge which each electron **feels** becomes **less** than Z = 2.

If you compute total energy, by substituting Z_{1} = 2 instead of 1.687, you will find the energy becomes higher (= **-74.83 eV** ) than -77.49 eV. ( See also this page. )

This is the same pattern as Fig.7.

**True** central charge always becomes less than "2" due to the existence of another electron.

If you adopt **wrong** charge parameter inside wavefunction, this energy becomes higher due to the increased kinetic energy.

*(Fig.12) Probability density of each electrons is changing depending on r _{12} ? *

As shown in Eq.17, the variational method **cannot** give correct ground state energy of helium.

Because Schrodinger's wavefunction **cannot** treat two electrons, which are **moving** around to **avoid** each other.

So in 1928-1930, Hylleraas used the special wavefunctions containing the interelectronic distance r_{12}, as shown in Fig.12.

This wavefunction becomes **larger**, as this interelectronic distance r_{12} is **longer**.

So, as you notice, this Hylleraas wavefunction of Fig.12 is **NOT** stationary.

This **Non-stationary** wavefunction clearly **contradicts** the probability concept of wavefunction.

*(Eq.18) *

The calculated result of Fig.12 becomes **-78.7 eV**, which is lower than -77.49 eV of Eq.17, but still higher than true experimental energy (= **-79.005 eV** ).

And depending on interelectronic distance r_{12}, the probability density at each point **changes**, which is very strange.

*(Eq.19) *

To get the ground state energy close to the experimental value, we have to use more than **1000** variational functions (= very **unrealistic** ! ) like Eq.19.

So these quantum mechanical methods have already reached their limits even in simple helium atom.

Furthermore, the latest computed value of -79.015 eV is **different** from the experimental value of -79.005 eV.

Because, wavefunction **cannot** consider correct nuclear motion (= reduced mass ) in helium atom.

*(Eq.20) Each electron = "superposition" of various wavefunctions ? *

Hylleraas fucntion containg interelectronic distance is too **complicated** to be applied to larger atoms.

So in larger atoms, they often use **configuration interaction** (= CI ) method.

In CI, atomic orbital is in the superposition state consisting of various wavefunctions ( with various principal quantum number ).

For example, in Eq.20, each electron's wavefunction consists of five different variational functions ( Clementi and Roetti ).

Surprisingly, these variational functions includes **unrealistic** charges (= 2.377, 4.396, 6.527 .. ).

Helium has Z=2 positive nuclear charge, so Z > 2 charges are **impossible**.

This is a fatal **defect** in unrealistic variational methods.

*(Eq.21) Combination of wavefunctions with different atomic radii in helium ( Z = 2 ).*

As a result, this helium wavefunction includes different combinations with **different** atomic **radii**.

For example, in Eq.21, atomic radius in Z_{1} = 1.417 is much larger than that in Z_{1} = 7.942.

( As I said. it is **impossible** that **Z=2** helium includes charge **Z _{1}=7.942** from the realistic viewpoint. )

The calculated result of Eq.20 becomes **-77.9 eV**, which is a little lower than -77.49 eV in Eq.17 (= not superposision ).

*(Fig.13) Two electrons avoid each other ?*

As a result, in the superposition of Eq.20, repulsive interaction between two electrons becomes **weaker** in some terms containing **different**-type wavefunctions.

This variational parameter of Z_{5} = 7.942 ( ← **unrealistic** ! ) is much larger than real helium nuclear charge Z=2, so the computed energy becomes higher only in each single wavfunction, as I said in Fig.7.

But due to the existence of terms consisting of **different-sized** orbitals, repulsive potential between two electrons becomes **lower**, which leads to the lower total energy.

This is the true mechanism in all of the present quantum chemistry ( ab-initio ).

As you feel, they **NEVER** admit each electron as clearly **separated** particle, these wavefunctions become extremely **complicated** and out of touch with reality. ( → Shut up and calculate ! )

*(Eq.22) Lithium ( Z = 3 ) Hamiltonian.*

Next we treat lithium atom (= Li ), which has Z = +3e positive nucleus, two 1s electrons, and one 2s electron.

So the Hamiltonian (= total energy E ) becomes as shown in Eq.22.

*(Eq.23) Slater type (= hydrogen ) orbitals.*

1s and 2s wavefunctions originate in hydrogen atom.

Z_{1} and Z_{2} are variational charge parameters, so this case in Li is also the **same** as Fig.7 and Fig.8.

*(Eq.24) If all three electrons in Li are in 1s orbital ....*

The important point is that if we try to consider all three orbitals in Eq.24 are 1s wavefunctions, the calculated value of Li ground state energy becomes less than **-214.3 eV**, which is **lower** than the experimental value (= **-203.5 eV** )

So this result completely **violates** the variation principle, because the calculated value is already lower than the experimental value before we do variational methods.

As a result, the idea that all three Li electrons are in 1s state is completely **wrong**.

As you see the result of Eq.24, Li atom uses Pauli exclusion principle as strong repulsive force.

This repulsive force of Pauli exclusion principle is more than **10 eV**, which is much **bigger** than usual spin-spin interaction (= fine structure level, **0.00005 eV** ).

So Pauli exclusion principle can **NEVER** be caused by spin-spin interaction.

*(Eq.25) Pauli exclusion principle = determinant ??*

So they introduced mathematical determinant to describe Pauli exclusion principle.

In determinant, when two columns or lines are the same (= for example 1s = 2s ), this wavefunction becomes **zero**.

So 2s function of Eq.25 **cannot** be 1s, they insist.

But as you feel, this determinant just forbids 2s to be equal to 1s, "mathematically" ( **NOT** physically ).

And it says **nothing** about concrete mechanism about Pauli exclusion principle.

As shown on this page, even in relativistic quantum field theory, they only shows very **abstract** anticommutator, **NOT** showing any more concrete thing.

This **vague** and **unkown** Pauli exclusion principle (= anticommutator ) is the main reason why the present quantum field theory ( including condensed matter and particle physics ) is completely **useless**.

*(Eq.26)*

Using Eq.25 in Eq.22, the calculated value becomes **-201.2 eV**.

As shown on this page, our new Bohr model can give the ground state energy of **-203.033 eV**, which is closer to the experimental value of -203.5 eV.

As I said in Eq.25, spin-spin interaction is **too weak** to cause Pauli exclusion principle, overcoming Coulomb attractive force.

So there is **only one** thing left to explain this important repulsive force of Pauli exclusion.

As shown on this page, **de Broglie** wave interference is the main reason of this Pauli exclusion principle.

*(Eq.27) Hamiltonian of hydrogen molecule ion (H2+)*

Hydrogen molecule ion (H2+) consists of two +e nuclei and one electron.

Eq.27 is the Hamiltonian of this H2+.

Using 1s hydrogen wavefunction, H2+ wavefunction becomes

*(Eq.28) H2+ wavefunction ?*

Z_{1} is variation parameter, and considering symmetry, wavefunction ψ is supposed to be the sum of wavefunctions based on nuclei 1 and 2.

Using Eq.28 in Eq.27, we get

*(Eq.29) Internuclear distance (= R _{e} ), dissociation energy (= D_{e} ).*

When variational parameter is Z_{1} is 1.24, internuclear distance (= R_{e}) between n1 and n2 is **1.06 Å**.

And dissociation energy (= D_{e} ) is **2.25 eV**.

Eq.29 result is a little different from the experimetal value of

*(Eq.30) Experimental value of H2+*

As I said in Fig.7, 1s wavefunction always keeps 1 × de Broglie wavelength.

And when the variational parameter Z_{1} is just equal to the **true** charge which the electron **feels**, the calculated energy becomes the lowest.

( Z_{1} = **1.24** is a little bigger than 1, and less than 2. which is good. )

*(Fig.7) When variational parameter Z _{1} = Z, the energy E is the lowest and stable.*

So H2+ ion by quantum chemistry also depends on an interger times de Broglie waves, which is the **same** mechanism as Bohr Sommerfeld model.

The result of Eq.29 is still different from the experimental value of Eq.30, so they **combine** 1s and 2s hydrogen wavefunctions like Eq.31. ( Dickinson. )

*(Eq.31) H2+ ion wavefunction ?*

where

*(Eq.32)*

Using Eq.31 and Eq.32 in Eq.27, the calculated result becomes

*(Eq.33) H2+ consists of two protons → This charge Z _{2} is correct ?*

When the charge parameters are Eq.33, the energy becomes lowest ( dissociation energy = 2.73 eV ), which is closer to the experimental value.

But the important point is that charge parameter Z_{2} included in 2p variational function becomes **unrealistic** value (= **+2.965 e** ).

H2+ molcule ion has only two positive nuclei (= 2 × +e ), so this charge of +2.965 e is **impossible**, and shows quantum chemistry **loses** touch with reality.

Of course, as shown in Fig.8, 2p wavefunction always keeps 2 × de Brolgie wave, and gives the lowest energy, when Z_{2} is real charge ( < 2 ).

But in this case of H2+ ion, this 2p function needs to **extend** into another nucleus, and be as close to both nuclei as possible.

When Z_{2} is lower than real value of "2", this wavfunction is far away from another nucleus.

( But of course, **Z _{2} > 2** is

*(Fig.14) H2+ wavefunction can be symmetric and antisymmetric ??*

If you see some textbooks, you will find H2+ ion wavefunction consists of symmetric (= Fig.14 upper ) and antisymmetric (= Fig.14 lower ) wavefunctions.

But from the realistic viewpoint, antisymmetric wavefunction is just mathematical tool, and **unreal**.

*(Fig.15) Symmetric wavefunction.*

When H2+ wavefunction is symmetric, variation parameter Z_{1} is close to 2, when
two nuclei are close to each other.

*(Fig.16) Antisymmetric wavefunction is unreal.*

On the other hand, in case of antisymmetric wavefunction, this charge variation parameter Z_{1} becomes **0.4**, when two nuclei stick to each other.

This is clearly **unrealistic**.

This antisymmetric wavefunction is often used in configuration interaction, because it can express **virtual** repulsive force among electrons.

So the present quantum chemistry is just mathematical thing.

In fact, when you ask if these wavefunctions are true or not, they only say "Shut up and calculate !", and **don't** try to explain the **physical** meaning of them.

*(Eq.34) Hydrogen molecule (H2) Hamiltonian*

Hydrogen molecule consists of two +e nuclei and two electrons.

Valence bond ( VB = Heitler-London ) method can give more correct value than MO method.

So most textbooks often refer to this method.

In fact, this Heiter-London method **disobeys** Virial theorem.

The wavefunction of H2 is

*(Eq.35) H2 wavefunction by Heitler-London methods. *

Using Eq.35 in Eq.34, we get

*(Eq.36)*

Dissociation energy of Eq.36 is different from the experimental value of

*(Eq.37) Experimental values of H2. *

Because H2 molecule includes repulsive interaction between two electrons.

As I said in helium section, these stationary wavefunctions need to be **dynamic** for two electrons to avoid each other.

This dynamic probability density clearly **violates** the original meaning of quantum wavefunction.

*(Eq.38) Real hydrogen molecule ? = more than 100 terms.*

To get the exact values, we have to use very complicated wavefunctions including variables such as r_{12}.

" r_{12} " means the distance between two electrons, so the probability at which we find the electron 1 is **changing** depending on the position of electron 2.

As shown in this page, the idea that wavefunctions show probability density holds **only** for one-electron hydrogen atom.

*(Eq.39) Combining multiple excited wavefunctions ← useful as real sicence ?*

To overcome the deficiencies of the Hartree-Fock wave function ( ex. electron correlation ), configuration interaction ( CI ) was introduced.

All ab-initio methods in quantum chemistry are based on this CI.

Each configulation function (= CSF ) is a linear combination of Slater determinants.

*(Eq.40) Doubly excited determinants.*

CI uses **virtual** excited orbitals such as singly excited (= Eq.39 ), doubly excited (= Eq.40 ) and triply excited.

As you see, these functions are too **compilcated** and disgusting.

*(Eq.41) Electrons avoid each other ?*

Then, why CI can give the calculated results close to the experimental values ?

One reason is that they are completely based on the same mechanism of **an integer** times de Brolgie waves and variation through Slater type (= hydrogen atom ) orbitals, as I said in Fig.7 and Fig.8.

And when we combine various excited wavefunctions, two electrons with different orbitals can avoid each other.

By increasing various terms, CI can find proper wavefunctions, which can give the values close to experiment.

But as shown in Eq.21 (= He CI ) and Eq.33 (= H2+ CI ), the variational charge parameters (= Z_{2} ) of excited virtual orbitals tend to be **unrealistic**.

( In H2+ case, Z_{2} = 2.965 is **greter** than the sum of two +e nuclei. )

*(Eq.42) CI wavefunctions.*

And as shown in Eq.39, Eq.40, Eq.42, this CI determinants are **too complicated** to be applied to larger molecules.

So we have to seek more realistic and simpler methods to express repulsive electrons.

To do so, first, we **must** ask what the wavefunctions and Pauli exclusion principle really are.

*(Eq.43) STO and GTO functions*

Slater-type orbitals (= STO ) originate in hydrongen atom, which can give more accurate energies than
Gauss-type orbitals (= GTO ).

But GTO basis set can be computed more easily than STO.

*(Eq.44)*

To use GTO basis set, we have to express STO function as a linear combinations of GTO functions.

*(Eq.45)*

6-31 G method uses 6 × GTO in inner shell, and 3 × GTO in outer shell.

And outer shells contain two different orbitals in each wavefunction.

The variation parameters and coefficients of these wavefunctions are determined in advance to express the energies close to the experimental value.

So 6-31G also depends on the same mechanism of an integer times de Broglie wavelength and variation as CI.

*(Eq.46)*

CI method takes extremely much time, so much simpler MP method was introduced.

For example, MP2 ( or MBPT2 ) uses second-order perturbation of energy, as shown in Eq.46.

Virtual energies (= a, b ) are higher than the original energies (= i, j ), so this second-order perturbation always negative.

As a result, MP2 always gives lower ground state energy than Hartree-Fock.

The inportant point is that MP2 can give **wrong** ground state energy, which is **lower** than the true energy.

( Of course, if the calculated ground state energy is lower than the actual energy, this theory is wrong. )

*(Fig.17) 1s hydrogen wavefunction = 1 × de Broglie wavelength.*

If you check hydrogen wavefunction based on Schrodinger equation in some websites or textbooks, you will find 1s radial wavefunction like Fig.17.

rR_{12} shows **1/2** de Broglie wavelength (= Fig.17 right ), so one orbit ( ∞ → 0 → ∞ ) includes 1 × de Broglie wavelength in the radial direction.

*(Fig.18) 2s hydrogen wavefunction = 2 × de Broglie wavelength.*

In case of 2s wavefunction, it includes 1 × de Broglie wavelegnth (= one crest + one trough ) from 0 to ∞.

So one round orbit ( ∞ → 0 → ∞ ) contains 2 × de Broglie wavelegnth.

As shown on this page, hydrogen n=1 and n=2 orbits in Bohr model are 1 × and 2 × de Broglie wavelength, respectively.

So both Schrodinger and Bohr model hydrogens **satisfy** an integer times de Broglie waves,

*(Fig.18') Schrodinger equation for hydrogen atom.*

The reason why χ = rR satisfies an integer times de Broglie wavelength is shown in Schrodinger equation of Fig.18'.

When we replace radial function R by rR, radial equation just expresses de Broglie momentum relation.

*(Fig.19) Probability density in 6-s hydrogen wavefunction.*

Fig.19 shows the probability density (= a square of amplitude ) of 6s hydrogen wavefunction.

It is known that the number of nodes becomes ( **n - l** ) in probability functions.

( "n" is the principal quantum number, and "l" is the angular momentum. ).

So in 6s probability wavefunction ( n = 6, l = 0 ), the number of nodes is 6 - 0 = **6**.

As a result, the wave amplitude (= rR_{6s} ) consists of 3 × crest and 3 × trough, which means 3 × de Broglie wavelength from 0 to ∞.

This means one orbit ( ∞ → 0 → ∞ ) contains **6 ×** de Brolgie wavelength, which number is just **equal** to Bohr model.

*(Eq.45) Sommerfeld quantization condition (A) + de Broglie relation (B).*

Eq.45 shows Bohr-Sommerfeld quantization condition.

In Eq.45A, "p" denotes angular momentum, and "p_{r}" denotes the radial momentum.

Eq.45B show tangental (= upper ) and radial (= lower ) de Broglie wavelengths. (= λ )

*(Eq.46) Sommerfeld quantization = integer times de Broglie wavelength.*

Substituting de Broglie relation of Eq.45B into Sommerfeld conditions of Eq.45A, we find that Sommerfeld conditions mean an integer times de Broglie wavelength in **both** tangential and radial directions.

*(Fig.20) 2p radial wavefunction = 1 × de Broglie wavelegnth. Angular momentum = 1.*

As shown in Fig.20, 2p hydrogen wavefunction includes one crest, so one round orbit becomes 1 × de Broglie wavelegnth.

Though 2p is n=2 energy level, why 2p radial orbit contains only 1 × de Broglie wavelegnth ?

( Compare Fig.20 right with Fig.18 right )

Because 2p contains angular momentum ( **l = 1** ), so one round **tangential** orbit also contains 1 × de Broglie wavelegnth. ( phase returns by φ = 0 → 2π )

As a result, the sum of radial and tangential waves becomes "2", which is consistent with n = 2 energy levels.

*(Eq.47) Sommerfeld quantization = integer times de Broglie wavelength.*

Basically, angular momentum of Schrodinger equation is also quantized as an integer times ħ, which is the same as Bohr model.

So you can easily understant the tangential direction in Schrodinger's hydrogen also satisfies an integer times de Broglie wavelength.

The important point is that also in the radial direction ( of 1s, 2s, 2p .. ), an integer times de Broglie wavelength is satisfied in Schrodinger equation.

Radial de Broglie wavelength (= λ_{r} ) is expressed using radial momentum (= p_{r} ), as shown in Eq.47.

*(Eq.48) Sommerfeld model and 2p Schrodinger radial wavefunction.*

As shown in Eq.48, one round orbit means A (= aphelion ) → B (= perihelion ) → A (= aphelion ).

At points A and B of elliptical orbits, the radial momentum become **zero** ( p_{r} = 0 ).

So according to de Broglie relation of Eq.47, radial wavelength becomes **infinite** at these points ( λ_{r} → ∞ ).

In 2p radial function, at both ends ( r = 0 and r = ∞ ), the wave phases **stop**, which means de Broglie wavelength becomes infinite.

( = the slopes of wavefunction become zero at both ends. )

*(Eq.49) Bohr model and Schrodinger's hydrogen.*

Eq.49 upper is Bohr model equation, and Eq.49 lower is Schrodinger's Hamiltonian.

The momentum p is changed into space derivative in Schrodinger equation.

Both hydrogens use conpletely the **same** relation ( E = T + V ).

*(Eq.50) One orbit of elliptical orbit.*

As shown in Eq.50, one round orbit means A (= aphelion ) → B (= pelihelion ) → A (= aphelion ).

And in hydrogen atom, orbital shape is symmetric, so one orbit can be expressed as **2 × ( A → B )**.

In Eq.50, the radius at perihelion is **r _{1}**, and that at aphelion is

So radial coodinate r is

( Of course, in

*(Eq.51) One orbit (= left ) and a half orbit (= right ).*

When one orbit contains 1 × de Broglie wavelength, a half orbit ( A → B ) contains 0.5 × de Broglie wavelength.

*(Eq.52) Mometum p = first derivative ∇*

As I said in Eq.49, momentum p is replaced by a first **derivative** (= ∇ ) in Schrodinger equation.

*(Eq.53) de Broglie relation.*

At perihelion and aphelion, the radial momentums become **zero**.

In Schrodinger hydrogen, zero momentum means wavelength λ is **infinite**.

So if you find the ends where de Broglie wavelength is **infinite**, they are perihelion and aphelion in Schrodinger's hydrogen.

*(Eq.54) 2p radial wavefunction.*

In 2p radial wavefunction, de Broglie wavelength becomes infinite at r=0 and r=∞.

Because at these points, the first derivative (= momentum ) becomes zero. ( slope is zero ).

*(Eq.55) *

As shown in Eq.55, as wavelength λ is closer to infinity, the phase change becomes **zero** (= Eq.55 right ).

*(Eq.56) A first derivative = momentum p.*

Eq.56 shows a first derivative of 2p wavefunctoin with respect to r, which means radial momentum p_{r}.

As shonw in Eq.56 and Eq.57, this derivative becomes **zero** at both ends of r=0 and r=∞.

*(Eq.57) *

Eq.57 means a half orbit ( from perihelion to aphelion ) in Schrodinger's hydrogen is from r=0 and r=∞.

As shown in Eq.54, this area contains 0.5 × de Broglie wavelength.

So one round orbit is 1 × de Brolgie wavelength in the radial direction.

*(Eq.58) *

The important point is that at the point of momentum p = 0, kinetic energy (= p^{2} ) doesn't become zero in Schrodinger equation.

This is very strange, because when p=0, p^{2} is NOT zero.

As shonw in Eq.58, the points where p^{2} (= second derivative ) is zero, are NOT equal to r =0 and r = ∞.

*(Eq.59) *

Of course, Bohr Sommerfeld quantization conditions of Eq.47 uses **first**-order momentum p.

So, the **first** derivative of wavefunction needs to be considered in this condition.

As a result, the radial ends in this condition become r = 0 and r= ∞.

*(Fig.21) Schrodinger's 2P "radial" wave function.*

As shown in Fig.21, the 2P "radial" wave function ( χ = rR_{21} ) contains the regions of the **negative** kinetic energy ( r < a1, a2 < r ).

When the r is bigger than a2, the Coulomb potential energy becomes *higher* than some maximum value.

In this region, to keep the total energy (E < 0) constant, the "radial" kinetic energy must be **negative** ( T < 0 ) !

Also in the region less than a1, radial kinetic energy becomes **negative**, though tangential kinetic energy (= angular momentum ) remains to be **positive**.

This is clearly **self-contradiction**.

So Schrodinger's hydrogen is wrong, and Bohr-Sommerfeld model is **right**.

2013/11/20 updated. Feel free to link to this site.