(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.
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 cannot 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.
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 (= ΨCID ) 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 Ψ0 ( this p.3-4 ).
Another Slater determinant includes two virtual excited states (= ψa, ψb ) replacing two original ground-state wavefunctions (= ψk, ψl ) = Ψab ( this p.8 ).
↑ two original ground state wavefunctions of ψk, ψl → two virtual excited-state wavefunctions of ψa, ψb
All wavefunctions (= ψi ) are supposed to be orthonormal ( this-p.5-(9) ), which means any overlap integrals of two different wavefunctions become zero (= ∫ ψmψn 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 (= ∫ Ψ0Ψab = 0, this-lower ), because the equation of ∫ Ψ0Ψab always includes the orthogonal relation between ground-state and excited wavefunctions (= ∫ψaψk = ∫ψbψl = 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 (= rij is the distance between two electrons i and j ) between two CI wavefunctions (= ΨCID ).
E = ∫ΨCID H ΨCID ← H = kinetic + Coulomb energies
Due to the orthogonal relation between ground-state and excited state wavefunctions (= ∫ψaψi = 0 ∫ψbψj = 0 ), the energy ② of the upper figure gives only unphysical exchange energy between two ground-state and excited-state energies (= ∫ψa(r1)ψb(r2)1/r12 ψk(r1)ψl(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 = H0 + H'
H0 = kinetic energy + Coulomb potential between electrons and nuclei
H' = Coulomb repulsive energy between two electrons
∫Ψ0 H0 Ψab = ∫ ψi(1) H0 ψa(1) ψj(2) ψb(2) = 0 ← ∫ ψj(2) ψb(2) = 0
∫Ψ0 H' Ψab = ∫ ψi(1) ψa(1) 1/r12 ψj(2) ψb(2) is not zero.
Basically, all excited energy equations except two-excited state (= ∫Ψ0 H Ψab = nonzero ) are zero like ∫Ψ0 H Ψa = 0 (= one-excited state ), ∫Ψ0 H Ψabc = 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 aab is far smaller than a0 ( aabaab << a0aab ).
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 ∫ ψi(1) ψa(1) 1/r12 ψj(2) ψb(2)
There is No ordinary Coulomb repulsive energy of ∫ ψi(1) ψi(1) 1/r12 ψa(2) ψa(2)
And as shown in the normalization of CI wavefunction (= ∫ΨCIDΨCID = 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 ∫Ψ0Ψab = 0 but ∫Ψ0 H Ψab 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 ∫Ψ0Ψab = 0 ), but only the numerator includes this "ghost" exchange energy between four different electron wavefunctions mixing two ground-states and two excited-states (= ∫Ψ0 H' Ψab is not zero ).
↑ This paradoxical "ghost-like" exchange energy (= ∫ ψi(1) ψa(1) 1/r12 ψj(2) ψb(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 (= rij ) like Hylleraas wavefunctions.
↑ The variable of interelectronic distance rij 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 rij 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 rij variable, hence wrong (= if the wavefunction includes this interelectronic distance rij, 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 = ∫Ψ0HΨab (← Not a normal ∫Ψ0HΨ0 or ∫ΨabHΨab, this p.18-(3.15), this p.29-30-2.3.1 ), 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 (= ∫ψaψb1/r12 ψkψl, 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 lower total energy 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 (= EMP2 ) is always lower than Hartree-Fock energy (= EHF ) 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 ( 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 cannot 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 overestimation 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 overcorrecting 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 (= rij ) required for true wavefunctions.
Trial wavefunctions or basis sets including the variables of interelectronic distance ( like r12 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.
