- Configuration interaction (= full-CI ) is unreal
- Coupled-cluster (= CCSD ) uses wrong tricks
- Møller–Plesset perturbation theory (MP) is false.
- Variational methods, Hylleraas wavefunctions give wrong energies.

*(Fig.1) CI method relies on unphysical negative exchange energy between virtual excited and ground-state electrons to illegitimately lower total energy.*

Quantum mechanical Schrödinger equations for multi-electron atoms ( such as helium ) and hydrogen molecule ion (= H+ ) are unable to have true solutions or true wavefunctions to conserve the constant total ( ground-state ) energy E (in any different electrons' positions ), so quantum mechanical Schrödinger equations and wavefunctions proved to be **false** ( this p.2-upper, this p.7 ).

Quantum mechanics just artificially chooses **fake** trial wavefunction or basis sets for unsolvable Schrödinger equations and manipulates various free parameters, which ad-hoc approximation is called variational methods ( this p.25, this p.1-right-first-paragraph ).

↑ They claim some infinite terms manipulating infinite parameters could become exact solutions of Schrödinger equations for multi-electron atoms, but this is untrue (= finding exact wavefunctions canceling multiple Coulomb potential energy terms to obtain the conserved total energy E is impossible ).

And there is **No** guarantee that infinite terms of chosen trial wavefunctions cannot get lower than true ground state energies violating the principle of variational methods. ← Quantum mechanical variational method claims artificially-chosen ( fake ) trial wavefunctions always give energies higher than true atomic ground-state energy, which actually has **No** guarantee.

Actually, physicists deliberately choose and fix exponent parameters to prevent the calculated total energies from lowering than true energies (= which means quantum mechanical variational methods are **false** ).

Of course, this quantum mechanical variational method just artificially choosing (fake) trial wavefunctions can **Not** predict exact atomic energies (= physicists can**not** know true atomic energies, until they compare the calculated values with experimental energies, this p.2-2nd-last-paragraph ). ← As a result, there is No such thing as quantum mechanical prediction, contrary to a lot of hypes.

It is far better to use experimental values such as atomic energies, shapes, properties from the beginning than to waste too much time in meaningless, time-consuming quantum mechanical equation such as Hartree-Fock, CI, CC, DFT...

Quantum mechanical ( artificially-chosen ) wavefunctions obeying Pauli principle must be expressed as (unphysical) antisymmetric wavefunctions or Slater determinants, whose Schrodinger energy calculation is called Hartree-Fock (= HF ) equations ( this p.6 ), which actually cannot describe multiple-electron atoms mixing the singlet and triplet wavefunctions, so quantum mechanical Pauli principle is false.

Furthermore these ordinary Hartree-Fock (= HF ) or molecular orbital methods are known to be unable to give the total energies close to experimental values even in their chosen trial wavefunctions. ← the difference between this Hartree-Fock equation and true energy is called correlation energy.

The current most-widely used quantum mechanical methods to seemingly get this correlation energy beyond Hartree-Fock is one-pseudo-electron density functional theory (= DFT ) that just artificially chooses fictitious exchange-correlation energy functional and pseudo-potential, so DFT is **fake** ab-initio unable to predict any values.

To get correlation energies or lower total energies toward true ground state energies by ordinary multi-electron methods (= not one-pseudo-electron DFT ), physicists introduced the very time-consuming impractical method called (full) configuration-interaction (= CI or full CI ) to lower the total energies than Hartree-Fock's wrong energy.

This "correlation energy" allegedly obtained by this CI method is based on **unrealistic** ad-hoc two electrons' exchange energies, which artificial trick can **Not** validate quantum mechanical ability of prediction (= As I said, all quantum mechanical methods including this full CI cannot solve Schrodinger equations nor predict exact energies, until physicists compare the calculated values with experimental values ).

Quantum mechanical wavefunctions must be expressed as (unphysical) Slater determinants or antisymmetric wavefunctions by considering Pauli principle (= exchanging two electron's variables must flip the sign of the total wavefunctions, this p.20-21 ).

The wavefunction (= Ψ_{CID} ) of configuration interaction (= CI or full-CI ) is expressed as the linear combination of more than one Slater determinants, as shown in the above figure.

One Slater determinant is the ordinary Hartree-Fock ground-state antisymmetric wavefunction Ψ_{0} ( this p.3-4 ).

Another Slater determinant includes two virtual excited states (= ψ_{a}, ψ_{b} ) replacing two original ground-state wavefunctions (= ψ_{k}, ψ_{l} ) = Ψ^{ab} ( this p.8 ).

↑ two original ground state wavefunctions of ψ_{k}, ψ_{l} → two virtual excited-state wavefunctions of ψ_{a}, ψ_{b}

All wavefunctions (= ψ_{i} ) are supposed to be orthonormal ( this-p.5-(9) ), which means any overlap integrals of two different wavefunctions become zero (= ∫ ψ_{m}ψ_{n} is zero, when m is Not equal to n, The case of non-orthogonal wavefunctions which also use unphysical exchange energy trick was explained later ).

As a result, the overlap integral between these two different Slater determinants also becomes zero (= ∫ Ψ_{0}Ψ^{ab} = 0, this-lower ), because the equation of ∫ Ψ_{0}Ψ^{ab} always includes the orthogonal relation between ground-state and excited wavefunctions (= ∫ψ_{a}ψ_{k} = ∫ψ_{b}ψ_{l} = 0 )

*(Fig.2) Unrealistic exchange energy between two ground-state and virtual excited-state electrons gives negative energy in configuration interaction (= CI ) trick.*

The total energy E is computed by putting Hamiltonian (= H ) energy consisting of electron's kinetic energy (= derivative ∇^{2} ), Coulomb attractive energy between electrons and nuclei (= Z ), repulsive interelectronic energy (= r_{ij} is the distance between two electrons i and j ) between two CI wavefunctions (= Ψ_{CID} ).

E = ∫Ψ_{CID} H Ψ_{CID} ← H = kinetic + Coulomb energies

Due to the orthogonal relation between ground-state and excited state wavefunctions (= ∫ψ_{a}ψ_{i} = 0 ∫ψ_{b}ψ_{j} = 0 ), the energy ② of the upper figure gives only unphysical electron-electron exchange energy between two ground-state and excited-state energies (= ∫ψ_{a}(r1)ψ_{b}(r2)1/r_{12} ψ_{k}(r1)ψ_{l}(r2) ← four different ground-state and excited-state wavefunctions are mixed ).

Other kinetic energies and electron-nuclear energies become zero (= orthogonal overlap integral is zero, this p.1,p.7, this p.8, this p.3-(3) ).

H = H_{0} + H'

H_{0} = kinetic energy + Coulomb potential between electrons and nuclei

H' = Coulomb repulsive energy between two electrons

∫Ψ_{0} H_{0} Ψ^{ab}
= ∫ ψ_{i}(1) H_{0} ψ_{a}(1) ψ_{j}(2) ψ_{b}(2) = 0 ← ∫ ψ_{j}(2) ψ_{b}(2) = 0

∫Ψ_{0} H' Ψ^{ab}
= ∫ ψ_{i}(1) ψ_{a}(1) 1/r_{12} ψ_{j}(2) ψ_{b}(2) is not zero.

Basically, all excited energy equations except two-excited state (= ∫Ψ_{0} H Ψ^{ab} = nonzero ) are zero like ∫Ψ_{0} H Ψ^{a} = 0 (= one-excited state ), ∫Ψ_{0} H Ψ^{abc} = 0 (= three-excited states ) ← the overlap (exchange) integrals are zero due to the orthogonal wavefunctions ( this p.30-37, this p.10-15 ).

③ of the upper figure is the non-zero excited energy, but this excited energy is negligibly smaller than the middle ② strange exchange energy, because the coefficient a^{ab} is far smaller than a_{0} ( a^{ab}a^{ab} << a_{0}a^{ab} ).

As a result, by manipulating each coefficient a, the middle ② of strange exchange energy between two electrons (= mixing four different ground and excited states wavefunctions ) can lower the total energy, which is the trick of configuration interaction (= CI ).

But this middle ② exchange energy is **unrealistic**, because there is No ordinary Coulomb potential energy or Coulomb integral in this strange exchange energy between two electrons.

There is only strange exchange energy of
∫ ψ_{i}(1) ψ_{a}(1) 1/r_{12} ψ_{j}(2) ψ_{b}(2)

There is No ordinary Coulomb repulsive energy of
∫ ψ_{i}(1) ψ_{i}(1) 1/r_{12} ψ_{a}(2) ψ_{a}(2)

And as shown in the normalization of CI wavefunction (= ∫Ψ_{CID}Ψ_{CID} = 1 ), this middle ② exchange energy is originally zero due to the orthogonal relationship (= overlap integral between ground-state and excited-state wavefunctions always becomes zero ∫Ψ_{0}Ψ^{ab} = 0 but ∫Ψ_{0} H Ψ^{ab} is Not zero ← this is trick of CI ).

So the denominator (= normalization ) of CI energy equation does Not include this ② exchange energy (= due to ∫Ψ_{0}Ψ^{ab} = 0 ), but only the numerator includes this "ghost" exchange energy between four different electron wavefunctions mixing two ground-states and two excited-states (= ∫Ψ_{0} H' Ψ^{ab} is not zero ).

↑ This paradoxical "ghost-like" exchange energy (= ∫ ψ_{i}(1) ψ_{a}(1) 1/r_{12} ψ_{j}(2) ψ_{b}(2) = trick of lowering CI energy ) between two electrons of ground-state and excited state can give **attractive** potential energy (= strangely attractive negative Coulomb energy between two virtual electrons ) contradicting the fact that Coulomb energy between two electrons must be repulsive and positive energy.

In case of non-orthogonal, the middle ② exchange energy can also contain ordinary exchange energy of illegitimately lowering kinetic energy (= which does not become zero when two wavefunctions are non-orthogonal ).

This paradoxical unphysical exchange energy between ground-state and excited-state lowering total energy illegitimately is the trick of configuration interaction's (= CI ) negative correlation energy.

The wavefunctions of this configuration interaction (= CI ) do Not include the variables of interelectronic distance (= r_{ij} ) like Hylleraas wavefunctions.

↑ The variable of interelectronic distance r_{ij} is indispensable for "true wavefunctions" that could cancel Coulomb potential energy between two electrons to obtain the conserved constant total energy.

So all CI wavefunctions that do Not include this r_{ij} interelectronic distance are unreal **wrong** wavefunctions.

Using arbitrary numbers of virtual excited wavefunctions and strange exchange energies, CI could lower the calculated total energy than true ground state energies

So Configuration interaction (= CI ) has to artificially adjust and fix the exponent parameters (= η of exponential function e^{-η r } ) at some values to prevent the calculated values from lowering than the true ground state energies ( this p.6-last-paragraph, p.16-Table.5 fixed the CI exponent ζ at an integer 32.0, this p.6-lower-optimized exponents ). ← This is Not true variational method, so false.

In fact, quantum mechanical Pauli repulsion cannot describe wavefunctions with more than three electrons.

So quantum mechanical wavefunctions with more than two electrons cannot contain this interelectronic r_{ij} variable, hence **wrong** (= if the wavefunction includes this interelectronic distance r_{ij}, the normal Hatree-Fock one electron approximation cannot be used, and the spatial wavefunction must consist of symmetric and antisymmetric variables, which is impossible ).

*(Fig.3) Coupled cluster (= CC ), which is Not variational, can give wrong energies lower than true ground state energy, so CC is Not a right theory.*

Instead of impractical time-consuming configuration interaction (= CI ), coupled cluster (= CC or CCSD ) theory is often used.

The point is this coupled cluster (= CC ) theory **contradicts** the original Schrödinger equation, and can give **wrong** energies lower than the true ground state energies, which means CC is intrinsically a **false** theory (= CC is Not variational method ).

This-(6.30) says

"The result is a set of equations which yield an energy that is **not** necessarily variational (i.e., may **Not** be above the true energy)" ← Coupled cluster (= CC ) is **false**.

This p.1-left-1st-paragraph says

"it is worth noting that
CCSD (= coupled-cluster with singles and doubles ) is **Not** variational, and it is also **Not** guaranteed
to give real-valued energies"

This p.16-top says

"Consequently, the total CC energy is **Not** necessarily above the
exact ground-state total energy, just as in perturbation theory"

In CC, total energy (= E ) is given in a **wrong** way putting Hamiltonian energy equation (= H ) between two different wavefunctions like E = ∫Ψ_{0}HΨ^{ab} (← Not a normal ∫Ψ_{0}HΨ_{0} or ∫Ψ^{ab}HΨ^{ab}, this p.18-(3.15), this p.27-2.29, p.29-2.40 ), which is the trick of CC lowering the total energy illegitimately faster than CI.

As shown in the above figure, the total energy of CC can be lowered by the unphysically-negative exchange energy between two electrons of ground state and excited state (= ∫ψ_{a}ψ_{b}1/r_{12} ψ_{k}ψ_{l}, this p.8, this-p.7-lower, this-lower, this-(4) ) like CI.

↑ This means CC contains only ① and ② energy parts of CI, and deliberately omits the annoying positive excited energy of ③ to illegitimately lower the total energy than CI.

So CC is known to be able to get total energy **lower than** true ground state energy (= CC is **Not** variational ), which fact **invalidates**
CC method.

There are many cases where coupled cluster (= CC, CCSD, CCSDT ) **disagreed** with experimental results.

This p.5-3rd-paragraph says

"The CCSD model systematically **overestimates** all core excitation energies".

This p.36-3rd-paragraph says

"CC **fails** for systems
with strong static correlation"

This-abstract-last says

"CCSD(T) **overestimates** the formation energy"

This p.6-left-2nd-paragraph says

"CCSDT we observe partially strong **overestimation** of the
total energy"

*(Fig.4) MP2 is not variational, because it can lower total energy than true energy illegitimately.*

Instead of extremely-time-consuming configuration interaction (= CI ) and coupled cluster (= CC ), Møller–Plesset perturbation theory (= MP) is often used to lower the total energy than Hartree-Fock (= HF ) energy using deceitful tricks.

As shown in above figure, MP2 total energy (= E_{MP2} ) is **always lower** than Hartree-Fock energy (= E_{HF} ) due to the artificially-introduced second term that is always **negative**.

The problem is this MP or MP2, which perturbation method (= PT ) is Not variational theory, often gives wrong energies lower than true ground state energies (= overestimation of energyes, this abstract-last, this p.7-2nd-paragraph
), which means Møller–Plesset perturbation theory is an **unreal** theory.

This p.12-last says

"the MPn methodology is **Not** variational – it is possible that the
MP2 estimate for the correlation energy will be **too large** instead of too small."

This p.10-last says

"MP theory uses a truncated Hamiltonian, and is therefore **Not**
variational. That means that one can**not** use the calculated energy as a measure
of the quality of the calculation."

This introduction-2nd-paragraph says

"Despite this popularity, the limitations of MP2-like methods are also well-known. In particular, they **fail** spectacularly for strongly correlated or metallic systems.9–11 Furthermore, a strong **overestimation** of dispersion interactions is observed for large polarizable systems"

This 1st-paragraph says

"it suffers from the usual MP2 **over**estimation of dispersion."

This p.2-2nd-last-paragraph says

"Thus, the value of E(2), the first perturbation to the Hartree-Fock energy, will always be negative. Lowering the energy is what the exact correction should
do, although the Møller-Plesset perturbation theory correction is capable of **over**correcting it, since it is **not** variational"

↑ As shown here, Møller–Plesset perturbation theory (= MP2 ) is a **defective** illegitimate theory.

*(Fig.5) Physicists artificially manipulate exponent parameter k to obtain experimental energies, which is Not true variational, so false.*

All CI, CC, MP2 are **false**, because these wavefunctions do **Not** include the interelectronic distance (= r_{ij} ) required for true wavefunctions.

Trial wavefunctions or basis sets including the variables of interelectronic distance ( like r_{12} which means the distance between electron-1 and electron-2 ) are called Hylleraas wavefunctions, which can give lower energies than full configuration interaction (= CI ).

↑ In fact, this interelectronic distance variable can Not be used in any atoms or molecules with more than two electrons due to paradoxical quantum mechanical Pauli principle's inability to mix singlet and triplet in one wavefunction ( this p.12, this p.3-(6)-Lithium-Hylleraas function uses the **wrong** asymmetrical wavefunctions contradicting the original singlet-triplet forms ).

If quantum mechanical Schrödinger equations or variational methods can give fake energies lower than true ground-state energies, quantum mechanics is proved to be **false** (= because quantum mechanical variational methods are said to always give values higher than true ground state energies = upper bound ).

In fact, in all these quantum mechanical variational methods, physicists must deliberately manipulate the (orbital) exponent parameters k, ζ, ξ, η of e^{-kr}, e^{-ζ r}, e^{-ξr}, e^{-ηr}.. to seemingly obtain (fake) total energy close to experimental value ( this p.6-7 ).

↑ They intentionally **stop** using this exponent parameter k as rigorous variational parameter giving the lowest energy, because if they try to find the exponent parameter k giving the lowest energy, this can be **wrong**, lower than true ground-state energies ( this p.6-IV says The exponents are considered as **adjustable** parameters. ).

This p.1-(1)(2), p.3-Table.I says "where αi, βi and γi are complex parameters (= exponent parameters of helium Hylleraas wavefunctions ) generated in a **quasi-random** manner: (= adjusting parameters by quasi-random manner means they are **Not** variational parameters giving the lowest energies )"

This p.12-2nd-paragraph says

"Also, I know of no published theoretical attempts to explain the excellent convergence found by Korobov with his highly nonlinear **fitting** of the
trial wavefunction..
Korobov’s approach may be likened to the work of **fitting** the Hydrogen
radial wavefunction with a set of gaussians, using floating exponents.
This sounds plausible, but at present it is just more **handwaving** about flexibility."

This p.4-left-last-paragraph says "The orbital exponent **may** be treated as a nonlinear (= exponent ) variational parameter,
though this is **Not** strictly necessary (= they don't try to find the lowest energy by using exponents as strictly-variational parameters )"

p.5-(24), p.9-Table.V intentionally **fixed** one non-linear parameter β = 1 and another exponent parameter α = 2.20 (= if they more strictly vary these parameters, they could find **fake** energies **lower** than true ground state energies ).

Schwartz intentionally **fixed** an exponent parameter k to 3.5 (= k is **Not** used as a variational parameter finding the lowest energy, this-introduction-(3), p.1147-Results says the parameter k was **fixed** at the value 3.5 ).

Also in the recent calculation, he **fixed** (= instead of vary ) the exponent parameter k at **2.0**, which was used as an **adjustable** scale parameter (= preventing the calculated values from getting lower than true helium energy ) not as a true variational parameter to obtain helium ground state energy based on Hylleraas wavefunctions including logarithmic wavefunctions ( this p.2-(0.3) ).

Another research also **intentionally fixed** exponent parameters (= η, ζ ) of helium Hylleraas wavefunctions at 6.00 or 6.50 ( this-p.6-lower~p.7-Table.1 ).

Drake also treated the exponent nonlinear parameters as **freely optimized** scale factors ( this p.4-Table.IV ).

↑ It means quantum mechanical variational methods can get **wrong** energy values **lower** than true ground state energies (= such as helium ) if physicists use various forms of wavefunctions (= including interelectronic distance like Hylleraas ) and parameters.

So they intentionally **fixed** those (exponent) parameters to **avoid** lowering the calculated values below the true ground state energies. ← **fake** variational method

This p.1-introduction-4th-paragraph (and p.5-Table.1 ) says "However, according to recent experimental result (Bergeson et al., 1998), the results from the
techniques reported above **overestimated** the helium ground-state energy" ← Quantum mechanical variational methods turned out to be Not variational, so false.

As a result, quantum mechanics is not only unable to predict any multi-electron atomic energies (= just manipulate exponent parameters k ) but also inconsistent with physical reality and the original quantum mechanical principle, so quantum mechanics is wrong.

2024/3/24 updated. Feel free to link to this site.