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The Hartree–Fock (HF) approximation has been an important tool for quantum-chemical calculations since its earliest appearance in the late 1920s and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy; but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion, this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and, since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke’s atom (two electrons in a parabolic well) to test the conjecture in a nontrivial case and to parameterize the difference between density functional and HF quantities, but find no violations of the conjecture. 

Biological processes in Southern Ocean surface waters have widespread impacts on global productivity and oceanic CO₂storage. Here, we demonstrate that biological calcification in the Southern Ocean exerts a strong control on the global distribution of alkalinity. The signature of Southern Ocean calcification is evident in observations as a depletion of potential alkalinity within portions of Subantarctic Mode and Intermediate Water. Experiments with an ocean general circulation model indicate that calcification and subsequent sinking of biogenic carbonate in this region effectively transfers alkalinity between the upper and lower cells of the meridional overturning circulation. Southern Ocean calcification traps alkalinity in the deep ocean; decreasing calcification permits more alkalinity to leak out from the Southern Ocean, yielding increased alkalinity in the upper cell and low‐latitude surface waters. These processes have implications for atmosphere‐ocean partitioning of carbon. Reductions in Southern Ocean calcification increase the buffer capacity of surface waters globally, thereby enhancing the ocean's ability to absorb carbon from the atmosphere. This study highlights the critical role of Southern Ocean calcification in determining global alkalinity distributions, demonstrating that changes in this process have the potential for widespread consequences impacting air‐sea partitioning of CO₂.

 

Anthropogenic CO₂emissions are inundating the upper ocean, acidifying the water, and altering the habitat for marine phytoplankton. These changes are thought to be particularly influential for calcifying phytoplankton, namely, coccolithophores. Coccolithophores are widespread and account for a substantial portion of open ocean calcification; changes in their abundance, distribution, or level of calcification could have far‐reaching ecological and biogeochemical impacts. Here, we isolate the effects of increasing CO₂on coccolithophores using an explicit coccolithophore phytoplankton functional type parameterization in the Community Earth System Model. Coccolithophore growth and calcification are sensitive to changing aqueous CO₂. While holding circulation constant, we demonstrate that increasing CO₂concentrations cause coccolithophores in most areas to decrease calcium carbonate production relative to growth. However, several oceanic regions show large increases in calcification, such the North Atlantic, Western Pacific, and parts of the Southern Ocean, due to an alleviation of carbon limitation for coccolithophore growth. Global annual calcification is 6% higher under present‐day CO₂levels relative to preindustrial CO₂(1.5 compared to 1.4 Pg C/year). However, under 900 μatm CO₂, global annual calcification is 11% lower than under preindustrial CO₂levels (1.2 Pg C/year). Large portions of the ocean show greatly decreased coccolithophore calcification relative to growth, resulting in significant regional carbon export and air‐sea CO₂exchange feedbacks. Our study implies that coccolithophores become more abundant but less calcified as CO₂increases with a tipping point in global calcification (changing from increasing to decreasing calcification relative to preindustrial) at approximately ∼600 μatm CO₂.

 

Two additions impacting tables 3 and 4 in ref. [1] are presented in the following. No significant impact is found for other results or figures in ref. [1].