Top page ( correct Bohr model)

End of Schrodinger equation. (14/6/11)

Truth of the fine structure constant α.

What does Planck constant really mean ?

True mechanism of useless quantum chemistry (13/11/20)

- What is variation method ?
- Truth of variational method in quantum chemistry.
- Quantum mechanics can NEVER show reality.

It is said that Schrodinger equation of quantum mechanics can give very exact energy values of various atoms.

But we **cannot** solve the equations other than simple hydrogen-like atoms.

So we need to use some **approximation**.

In these approimations, variational methods can give most exact values about the energies of atoms.

*(Fig.1) Quantum mecanical helium becomes chaotic and unstable.*

But as you know, the quantum mechanical atoms can NEVER tell us **true** states of electron's motion.

A single electron is actually **quantized** particle with definite mass and charge -e according to experiments.

So it is **inconsistent** with the experimental results to insist one electron is spreading into **all space** as **clouds**.

Furthermore, if two electrons of stable helium has **NO** angular momentum (= 1s ), this helium model would be extremely **unstable**.

It means that quantum mechanical peturbation or variational methods **cannot** express **real** states of atoms at all.

*(Eq.1) Hamiltonian of helium atom.*

Helium-like atom has **two** nagative electrons ( 1 , 2 ), and **one** positive nucleus (= +Ze ).

( In neutral helium atom, this Z is "2". )

Two first terms of Hamiltonian in Eq.1 represent kinetic energies of electron 1 and 2.

And the last term of Eq.1 is **repulsive** Coulomb interaction between two electrons.

( r_{12} is the distance between two electrons. )

*(Eq. 2) "1s" wavefunction of hydrogen-like atoms.*

When we try to solve helium's Schrodinger equation, we use **1s** wavefunction of **hydrogen**-like atom.

( Z_{1} = 1 is hydrogen atom, and Z_{1} = 2 is helium ion. )

Using two wavefunctions of Eq.2, we express neutral helium wavefunction, as follows,

*(Eq. 3)*

Substituting Eq.1 and Eq.3 into the energy (= W ) equation of

*(Eq. 4)*

we obtain

*(Eq. 5)*

When Z = 2, the total energy W of helium becomes **-74.833** eV, which is different from the experimental value of **-79.0051** eV.

Changing the atomic number Z, we get ground state energies of various helium-like atoms (ions), as follows,

Atoms | Experiment (eV) | new Bohr (eV) | Perturbation (eV) | Bohr Error | Pert Error |
---|---|---|---|---|---|

He | -79.0051 | -79.0037 | -74.8330 | +0.001 | +4.172 |

Li+ | -198.093 | -198.984 | -193.885 | -0.89 | +4.21 |

Be2+ | -371.615 | -373.470 | -367.362 | -1.85 | +4.25 |

B3+ | -599.60 | -602.320 | -595.262 | -2.72 | +4.34 |

In Table 1, "perturbation" means quantum mechanical approximation and "new Bohr" means computed Bohr's orbits which are **perpendicular** to each other as shown in Top page or Fig.2.

*(Fig.2) New Bohr model Helium.*

New Bohr's helium model can give just **exact** ground state energy ( except only for small relativistic effect ).

And of course, Bohr's orbits can also give **concrete** electron's motion, and agree with de Broglie wave's nature and **stability** of helium.

So we can apply this model to 1s inner states of various other atoms.

As you notice, quantum mechanical pertubation methods show almost **same error** (= about +4 eV ), even when the atomic number is **bigger**.
This is **unnatural**.

*(Eq. 6) Variation functions.*

Next we try quantum mechanical **variational** methods of helium-like atoms.

Here we consider charge Z_{1} as variational **parameter** and change it.

Substituting Eq.6 into Eq.4,

*(Eq. 4)*

We obtain,

*(Eq. 7)*

In Eq.7, "Z" is atomic number included in Hamiltonian of Eq.1.

So this Z is **fixed** value, which is different from variational parameter Z_{1}.

We vary Z_{1} to **minimize** the total energy as follows,

*(Eq. 8)*

For example, in neutral helium ( Z = 2 ), when *Z _{1}* = Z -5/16 = 2-5/16 =

Substituting this Z_{1} = Z - 5/16 into Eq.7, the total energy W becomes,

*(Eq.9)*

For example, substituting Z = 2 and E_{1s} = -13.606 into Eq.9, the total energy (E) becomes **-77.49 eV**, which is closer to the experimental value (*-79.005 eV*) than the perturbation of Table 1.

But this is the **limit** of this method about two-electron atom.

Atoms | Experiment (eV) | new Bohr (eV) | Variation (eV) | Bohr Error | Vari Error |
---|---|---|---|---|---|

He | -79.0051 | -79.0037 | -77.4904 | +0.001 | +1.515 |

Li+ | -198.093 | -198.984 | -196.552 | -0.89 | +1.55 |

Be2+ | -371.615 | -373.470 | -370.019 | -1.85 | +1.60 |

B3+ | -599.60 | -602.320 | -597.919 | -2.72 | +1.68 |

As shown in Table 2, both new Bohr model and quantum mechanical variational method give good results.

But as you notice, variational methods **cannot** get the exact values especially in neutral **helium** and **lithium ion**.

And it is very **unnatural** that the error ( about +1.5 eV ) doesn't change much, even when the atomic number is bigger.

These results demonstrate that the quantum mechanical variational method does **NOT** express the **true** configuration of neutral helium.

( Of course, originally, quantum mechanical model has no reality. )

As the atomic number becomes **bigger**, true **errors** ( based on the **repulsive** interaction between two electrons ) are **hidden** under the influence by the **big** positive nucleus.

*(Eq.10) Quantum mechanical helium.*

To get closer to the experimental values, we heve to use more than a **thousand** variational terms like Eq.10.

And to express **repulsive** force between electrons, these terms need to include variable ( r_{12} ), which is the **distance** between electrons.

So these wavefunctions can **NOT** show real helium state at all.

The existence of r_{12} means the **probability** at which we find electron 1 in some place is **changing** with respect to the position of electron 2.

So Eq.10 is **NOT** constant probability density wave.

*(Eq.10') Electron clouds = signle electron is divided ?*

Quantum mechancs often insists that single electron of hydrogen ( or other atomes ) are **spreading** into all space as **electron clouds**.

But if a single electron is **actually** divided, these energy values are extremely **higher** due to new generated repulsive forces between fractional electrons.

In Eq.10' left, only attractive forces between nucleus and a single electron exists.

But if an single electron is divided into two fractional charges ( e- = -1/2 e -1/2 e ) as electron clouds, new **repulsive** forces are generated between them, though Coulomb attractive force remains the same.

This fact clearly indicates a single electron is **not** divided **inside** various atoms and molecules.

In this section, we explain why the above **unrealistic** variational methods can give values close to experimental values.

( See also true mechanism of useless quantum chemistry. )

*(Eq. 11) "1s" hydrogen wavefunction.*

In the above section, we use 1s hydrogen-like wavefunction to get helium ground state energy.

As you know, Bohr model is completely **equal** to quantum mechnics in the energy levels of hydrogen-like atom.

So 1s wavefunction includes the same informations about average electron **distribution** (= Bohr radius ) and average **kinetic energy** (= T ).

*(Fig.3) Bohr model = "1s" Wavefunction.*

Quantum mechanical hydrogen also satisfies **Virial theorem**, E = 1/2V = -T .

( See Virial theorem of Schrodinger's hydrogen. )

So the same energy E means the same average kinetic energy T and distribution (= Bohr radius / Z_{1} ).

So from here, instead of 1s wavefunction, we use Bohr's orbit to explain variational methods.

When the electron is moving around the *+Z* nucleus on the orbit of radius *r*, the potential energy (V) is,

*(Eq. 12)*

When Bohr's orbit is **1 × de Broglie wavelength**, the **average** radius r becomes.

*(Eq. 13)*

where a_{0} is Bohr radius.

As I said, Schrodinger's 1s wavefunction is also 1 × de Broglie wavelength around Ze+ nucleus.

See also this page. So the average potential energy (= distribution radius ) is the same as Bohr model.

According to Virial theorem, the total (E) and kinetic (T) energies at this average radius r become,

*(Eq. 14)*

where E = 1/2 V, and T = -E.

When the variational charge parameter **Z _{1}** of Fig.3

When the momentum operator acts on the wavefunction, it gives kinetic energy.

So the wavefunction itself **includes** the informations of kinetic energy (= momentum ).

As a result, the change of **wavefunction** causes the kinetic energy (T) **change**, as follows,

*(Eq. 16)*

where Eq.14 and Eq.15 are used.

Only wavefunction's charge Z_{1} varies, and the nuclear charge (Z) keeps constant.

So the average potential energy (V) at r_{1} becomes,

*(Eq. 17)*

As a rsult, from Eq.16 and Eq.17, the total energy ( E = T + V ) becomes,

*(Eq. 18)*

The partial differential of Eq.18 with respect to Z_{1} is

*(Eq. 19)*

This result shows that when **Z _{1} = Z**, the total energy (E

*(Eq. 19') Mechanism of variational methods.*

When Z_{1} becomes bigger than Z (= Eq.19' left ), the *increase in kinetic energy* (= T ) becomes bigger than the decrease in the potential energy (= V ). ( T is **a square** of Z_{1}, compare Eq.16 and Eq.17. )

And when Z_{1} becomes smaller than Z (= Eq.19' right ), the decrease in kinetic energy is smaller than the increase in potential energy. ( So when Z_{1} =Z, total energy E is the **lowest**. )

This is the true mechanism of variational methods.

Here we use the atomic number Z.

But when the system includes **several** nuclei and electrons, **average positive charges** including overall particles mean this Z.

*(Eq. 20) If we use 2s (or 2p) wavefunction ?*

Even if we use 2s (or 2p ) type wavefunction, this result is the same as 1s function.

The difference is the value Z_{1} becomes **twice** the real atomic charge Z.

( In this case, 2s, 2p wavefunctions also mean 1 × de Broglie wavelength. )

*(Eq.21) Hydrogen moelcule ion (H2+). *

where

*(Eq.22)*

So as shown on this page, hydrogen molecule ion (H2+) variational functions give **unrealistic** Z values.

In Eq.21 χ_{1} is the **1s** hydrogen atom wavefunction, and χ_{2} is **2p** H atom wavefunction.

σ, Z' and Z'' are variational parameters.

Calculation results of variational methods are

*(Eq.23) *

As you see in Eq.23, the positive charge of 2p wavefunction becomes **2.868**.

Of course, H2+ molecule ion has **only 2** nuclei, so Z'' = 2.868 is **impossible**.

They try to express **1 ×** de Broglie wavelenth using **2p** wavefunction, so the variational charge Z'' becomes **twice** the real value. ( See Eq.20. )

*(Fig.4) 2 × 1s wavefunction = helium.*

As shown in Table 2, the variational method can give similar results to new Bohr model.

Basically 1s hydrogen wavefunction is **uniformly** distributed around nucleus.

So also in variational helium model, two electrons are uniformly distributed around nucleus.

*(Fig.5) New Bohr model = two electrons are uniformly distributed.*

As shown on Top page, when two electrons' orbits are **perpendicular** to each other, it means two electrons are **uniformly** distributed around nucleus.

As a result, helium is **not** electrically poralized, and does NOT form compounds with other atoms or itself.

Due to the uniform distribution + variatinal charge Z (= average charge, see Eq.19 ), the variational method of Table 2 can give similar results to new Bohr model.

*(Fig.6) Weak points of variational methods = NO clear motions.*

As shown in Table 2, variational method using 2 × 1s wavefunction can get **-77.4904 eV**, which is a little different from the experimental values of **-79.0051 eV**.

On the other hand, new Bohr model can get the exact value **-79.0037 eV**, except for small relativistic effect.

The difference between -77.4904 eV and -79.0051 eV is very important.

Because this difference is clearly caused by the **replusive interaction** ( motion ) between two electons.

Schrodinger's wavefunction can not show clear electron's motion and **relative** positions of two electrons, so it **cannot** explain the exact effects caused by the repulsive interactions and **reduced mass**.

This is the limit of variational methods.

In the upper section, we have proved that the variational methods of helium is similar to new Bohr model helium.

How about the lithium atom ?

The ground state energy of the lithium atom is **-203.48 eV**.

The hamiltonian operator of the lithium is,

*(Eq.24)*

where Δ = ∇^{2}.

If we calculate the lithium energy like Eq.4 and Eq.6, the result becomes *-214.3 eV*, which is *lower* than the experimental value.

This is strange.

This strange result is caused by the fact that the third electron of the lithium *can not* enter the 1S orbit.

So we need to *restrict* the variational wavefunction to get the correct result.

In the variational methods of the lithium, we use the following third-order **determinant** as the variational wavefunction.

*(Eq. 25)*

where 1S() and 2S() are the hydrogen wavefunctions.

And the numbers of 1,2, and 3 mean the electron numbers.

Why do we use the **determinant** ?

In the determinant, when the two electrons of them are in the *same* state, the whole wavefunction becomes **zero**.

If the wavefunction becomes zero, the total energy becomes *zero*, which is **higher** than the ground state energy ( -203.48 eV ).

So we can't use the 1S wavefunction in the third electron of the lithium.

When we use Eq.25, and vary the variational parameters Z_{1} and Z_{2}, the minimum total energy of the lithium becomes **-201.2 eV**, which is close to the experimental value ( **-203.48 eV** ).

In this case, Z_{1} and Z_{2} are *+2.686* and *+1.776*, respectively.

Because the two 1S electrons are closer to the nucleus than the 2S electron.

Surprisingly, these approximate value close to the experimental one was gotten in **1933** by manual calculation of the variational methods.

(E.B. Wilson, Jr., J. Chem. Phys., 1, 210, 1933)

But of course, "real" Bohr model could not give this excellent value in 1933 **due to the lack of computers**.

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

Li | 1949.0 | 1.000000 | -203.033 | -203.480 | 0.47 |

Be+ | 1427.0 | 1.000000 | -388.785 | -389.826 | 1.04 |

B2+ | 1125.0 | 1.000000 | -635.965 | -637.531 | 1.56 |

C3+ | 928.0 | 1.000000 | -944.46 | -946.57 | 2.11 |

N4+ | 790.5 | 1.000000 | -1314.25 | -1317.01 | 2.76 |

O5+ | 688.0 | 1.000000 | -1745.70 | -1748.82 | 3.12 |

F6+ | 609.4 | 1.000000 | -2237.60 | -2242.21 | 4.61 |

Ne7+ | 546.0 | 1.000000 | -2791.15 | -2797.12 | 5.97 |

*(Fig.7) Three electron atomic model.*

As shown in Table 3 or this page , our new Bohr model lithium (= Fig.7) gives **-203.033eV**, which is *more correct*.

And other three-electron atoms (ions) can be explained using new Bohr's model correctly, too.

When the Z_{1} is bigger, and Z_{2} is smaller, the *distributions* of the 1S and 2S electrons are **more apart** from each other. (Fig.8)

*(Fig.8) The change of the probability density in 1S and 2S electrons.*

So in the lithium, the *overlapping* parts of the 1S and 2S wavefunctions become *small*, and *the loss by the antisymmetric property of the determinat becomes small, too*.

Of course, the distributions of the 1S (Eq.2) and 2S electrons (of hydrogen-like atoms) are the **same** as the Bohr model (= the potential energies are the same in Bohr model and the quantum mechanics.)

So also in the lithium, Bohr model is similar to the quantum mechanics.

But as shown on this page, mathematical ( not physical ) concepts such as wavefunctions and determimants clearly **obstructs** the development of science.

*(Fig.9) Superposition = unrealistic Many-worlds !*

As you know, quantum mechanics **cannot** describe the electron's concrete motion at all.

It only shows the probability density of the electron in hydrogen atom.

So to explain wavefunction collapse, we need **unrealistic** many worlds.

*(Fig.10) 1 × de Broglie wavelength = hydrogen ground state.*

As shown on this page, Schrodinger's hydrogen **also** satisfies the condition of an integer times de Broglie wavelength.

This is the reason why Schrodinger's hydrogen gives completely the same results as Bohr model.

As shown in Fig.10, in the ground state of hydrogen atom, Bohr model consists of **tangential** de Broglie wave, and Schrodinger's model ( rR ) consists of "**radial**" de Broglie wave.

( It is easily understood also by seeing the probability graphs of |rR|^{2}. )

Actually, the probability of ground state of hydrogen is the highest around **Bohr radius**.

The problem is that Schrodinger's radial wavefunction must be always from zero to **infinity**.

And the 1s de Broglie wave is **linear**, so the opposite phases of them overlap and **cancel** each other out.

*(Fig.11) Schrodinger's wavefunctions always spread to infinity.*

In various atoms and molecules such as helium and hydrogen molecule, Schrodinger's wavefunction **ALWAYS** spread to **infinity**.

This means we can find the **bound-state** electron near **infinity** even in very stable helium atom !

Of course, this is **impossible**, and quantum mechanics ( chemistry ) has NO ability to describe real states of various atoms and molecules.

Schrodinger's equation of helium atom cannot be solved, but variational functions of it are always spreading to infinity.

*(Fig.12) Schrodinger's 2P "radial" wave function (angular momentum = 1).*

As shown in Fig.12, the 2P "radial" wave function ( χ = rR_{21} ) contains the regions of the **minus** 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 minus !

Furthermore, in the region from 0 to a1 of Fig,12, the potential energy is **lower** than the total energy.

But the **tangential** kinetic energy increases much faster than others.

To cancel the increased **tangential** kinetic energy, radial kinetic energy has to be minus.

These unreasonable states clearly show Schrodinger's hydrogen atom is **wrong**.

*(Fig.13) Quantum mecanical helium becomes chaotic and unstable.*

In the quantum chemistry, if we use two 1s wavefunction of hydrogen atom, we can get the approximate gound state energy of helium.

But as you notice, 1s state has NO angular momentum.

In this state, two electrons of helium become chaotic and very **unstable** due to repulsive Coulomb force between them.

So the quantum mechanical helium is completely **different** from the actual helium atom.

*(Fig.14) Quantum mechanical helium.*

To get the exact ground state energy of helium (or hydrogen molecule ) using Schrodinger's equation, we have to use more than a **thousand** variational functions, as shown in Fig.14.

*(Fig.15) Probability density of helium or H2 ?*

And these variational functions must include the variables (= r_{12} ) which express the **distance** between two electrons.

So, different from static hydrogen atom, the probability density at which we find electron 1 of helium (or H2 ) is always **changing** depending on the position of electron 2.

This variational functions of helium clearly shows the two electrons of helium are actually **moving**, even in the vague probability density.

*(Fig.16) Reduced mass = electrons and nucleus are actually moving !*

As you know, if we use the reduced mass of electron, we can get **more exact** values of the energy levels even in hydrogen atom.

This fact clearly shows that the electron and nucleus are actually moving interacting with each other.

Of course, as fractional charges of electron have **not** been found, we have to express **real** motions of the two electrons in helium atom.

As shown on this page, even pilot wave (Bohm) theory **cannot** explain two-electron helium at all, in quantum mechanics.

*(Fig.17) Calculation results using reduced mass are more exact. *

The experimental value of ground state hydrogen is **-13.598** eV.

When we calculate this ground state energy using Schrodinger equation (or Bohr model ), it becomes **-13.606** eV.

If we replace the usual electron mass by **reduced mass** considering nuclear motion, this calculation result becomes more exact (= -13.598 eV ).

This fact clearly shows the bound-state electron (and nucleus ) is **actually moving**.

Also in helium ion ( He+ ), if we use reduced mass, we can get more exact result (= **-54.415** eV ) than the usual electron mass (= -54.423 eV ).

The experimental value of He+ is -54.417 eV.

*(Fig.18) Two-electron Atomic Model ( He, Li+, Be2+, B3+, C4+ ... )*

As shown on Top page, if the two de Broglie waves (= 1 × wavelength ) are overlapping each other, their opposite wave phases **cancel** each other due to **destructive** interference.

( These destructive interferences of electron's de Broglie waves were actually observed in Davisson and Germer experiments. )

So the two electrons's orbit have to be **perpendicular** to each other to avoid destructive inteference.

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 |

As shown in Table.4, these two-electron atomic model can completely explain actual energy states of various atoms.

The helium ground state energy ( -79.0037 eV ) is **more exact** than the latest quantum mechanical variational methods ( -79.015 eV ), because quantum mechanical helium **cannot** deal with nuclear motion ( reduced mass ) correctly.

About the computing methods and programs, see this page.

And this model can express three-electron atoms such as lithium correctly, too.

*(Fig.19) New Bohr's helium (= A.) is not electrically polarized.*

Furthermore, this helium model is just consistent with the fact that helium atom is most stable and **doesn't** form compounds with other atoms (or itself ).

Because when the two orbits are perpendicular to each other, the space around 2e+ nucleus becomes just **neutral** due to uniform distribution.

And of course, there is no space into which the third electron enter in Fig.18 helium, considering the **stability** of de Broglie waves (= Pauli exclusion principle ).

If the third electron enter them, **destructive** interferences of their de Broglie waves make them unstable.

*(Fig.20) "Mathematical" Schrodinger equation vs. "real" Bohr model.*

We **cannot** know what the Schrodinger's wavefunctions really are and just **give up** asking what they are.

Furthermore, as shown on this page, even if we study about relativistic quantum field theory and string theory, we **cannot** know what mysterious Pauli exclusion principle really is.

They just insist anticommutation of mathematical operators means Pauli exclusion principle.

So, as shown on this page, we **cannot** develop and try easier and useful methods in this very **limited** and vague conditions.

As this **useless** present quantum chemistry **cannot** be applied to various fields such as molecular biology and nanotechnology, their developments all **stop** now in the molecular levels. This is **serious** problem.

*(Fig.21) Density functional theory = approximation.*

First, density functional theory (= DFT ), which is often used now, is only **approximation**, and **NOT** ab-initio method.

DFT is one of **semi**-empirical methods.

We can **artificially choose** various approximations such as LDA to match experimental results.

So popular DFT itself is **NOT** a true theory.

2013/2/9 updated. Feel free to link to this site.